/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 149 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), 287 ms] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 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) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRelTRS (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxWeightedTrs (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTypedWeightedTrs (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (46) CdtProblem (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 30 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 15 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 24 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Types: f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f a :: a:h:cons_f h :: a:h:cons_f -> a:h:cons_f encArg :: a:h:cons_f -> a:h:cons_f cons_f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f encode_f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f encode_a :: a:h:cons_f encode_h :: a:h:cons_f -> a:h:cons_f hole_a:h:cons_f1_0 :: a:h:cons_f gen_a:h:cons_f2_0 :: Nat -> a:h: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: TRS: Rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Types: f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f a :: a:h:cons_f h :: a:h:cons_f -> a:h:cons_f encArg :: a:h:cons_f -> a:h:cons_f cons_f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f encode_f :: a:h:cons_f -> a:h:cons_f -> a:h:cons_f encode_a :: a:h:cons_f encode_h :: a:h:cons_f -> a:h:cons_f hole_a:h:cons_f1_0 :: a:h:cons_f gen_a:h:cons_f2_0 :: Nat -> a:h:cons_f Generator Equations: gen_a:h:cons_f2_0(0) <=> a gen_a:h:cons_f2_0(+(x, 1)) <=> h(gen_a:h: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:h:cons_f2_0(n9_0)) -> gen_a:h:cons_f2_0(n9_0), rt in Omega(0) Induction Base: encArg(gen_a:h:cons_f2_0(0)) ->_R^Omega(0) a Induction Step: encArg(gen_a:h:cons_f2_0(+(n9_0, 1))) ->_R^Omega(0) h(encArg(gen_a:h:cons_f2_0(n9_0))) ->_IH h(gen_a:h:cons_f2_0(c10_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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, c_f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: FULL ---------------------------------------- (19) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (20) 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(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a encArg(h(x_1)) -> h(encArg(x_1)) 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_h(x_1) -> h(encArg(x_1)) f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (21) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (22) 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(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) [1] encArg(a) -> a [0] encArg(h(x_1)) -> h(encArg(x_1)) [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_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (23) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, c_f(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) [1] encArg(a) -> a [0] encArg(h(x_1)) -> h(encArg(x_1)) [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_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] The TRS has the following type information: f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f c_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f a :: a:c_f:h:cons_f h :: a:c_f:h:cons_f -> a:c_f:h:cons_f encArg :: a:c_f:h:cons_f -> a:c_f:h:cons_f cons_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_a :: a:c_f:h:cons_f encode_h :: a:c_f:h:cons_f -> a:c_f:h:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (25) 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_h_1 f_2 Due to the following rules being added: encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] encode_h(v0) -> a [0] f(v0, v1) -> a [0] And the following fresh constants: none ---------------------------------------- (26) 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(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) [1] encArg(a) -> a [0] encArg(h(x_1)) -> h(encArg(x_1)) [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_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] encode_h(v0) -> a [0] f(v0, v1) -> a [0] The TRS has the following type information: f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f c_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f a :: a:c_f:h:cons_f h :: a:c_f:h:cons_f -> a:c_f:h:cons_f encArg :: a:c_f:h:cons_f -> a:c_f:h:cons_f cons_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_a :: a:c_f:h:cons_f encode_h :: a:c_f:h:cons_f -> a:c_f:h:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (27) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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(c_f(a, y'), c_f(a, y)) -> f(a, f(f(f(a, f(f(f(a, a), h(a)), y')), h(a)), y)) [2] f(x, c_f(a, y)) -> f(a, f(f(c_f(a, x), h(a)), y)) [1] f(x, c_f(a, y)) -> f(a, f(f(a, h(a)), y)) [1] encArg(a) -> a [0] encArg(h(x_1)) -> h(encArg(x_1)) [0] encArg(cons_f(a, a)) -> f(a, a) [0] encArg(cons_f(a, h(x_11))) -> f(a, h(encArg(x_11))) [0] encArg(cons_f(a, cons_f(x_12, x_2''))) -> f(a, f(encArg(x_12), encArg(x_2''))) [0] encArg(cons_f(a, x_2)) -> f(a, a) [0] encArg(cons_f(h(x_1'), a)) -> f(h(encArg(x_1')), a) [0] encArg(cons_f(h(x_1'), h(x_13))) -> f(h(encArg(x_1')), h(encArg(x_13))) [0] encArg(cons_f(h(x_1'), cons_f(x_14, x_21))) -> f(h(encArg(x_1')), f(encArg(x_14), encArg(x_21))) [0] encArg(cons_f(h(x_1'), x_2)) -> f(h(encArg(x_1')), a) [0] encArg(cons_f(cons_f(x_1'', x_2'), a)) -> f(f(encArg(x_1''), encArg(x_2')), a) [0] encArg(cons_f(cons_f(x_1'', x_2'), h(x_15))) -> f(f(encArg(x_1''), encArg(x_2')), h(encArg(x_15))) [0] encArg(cons_f(cons_f(x_1'', x_2'), cons_f(x_16, x_22))) -> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) [0] encArg(cons_f(cons_f(x_1'', x_2'), x_2)) -> f(f(encArg(x_1''), encArg(x_2')), a) [0] encArg(cons_f(x_1, a)) -> f(a, a) [0] encArg(cons_f(x_1, h(x_17))) -> f(a, h(encArg(x_17))) [0] encArg(cons_f(x_1, cons_f(x_18, x_23))) -> f(a, f(encArg(x_18), encArg(x_23))) [0] encArg(cons_f(x_1, x_2)) -> f(a, a) [0] encode_f(a, a) -> f(a, a) [0] encode_f(a, h(x_111)) -> f(a, h(encArg(x_111))) [0] encode_f(a, cons_f(x_112, x_25)) -> f(a, f(encArg(x_112), encArg(x_25))) [0] encode_f(a, x_2) -> f(a, a) [0] encode_f(h(x_19), a) -> f(h(encArg(x_19)), a) [0] encode_f(h(x_19), h(x_113)) -> f(h(encArg(x_19)), h(encArg(x_113))) [0] encode_f(h(x_19), cons_f(x_114, x_26)) -> f(h(encArg(x_19)), f(encArg(x_114), encArg(x_26))) [0] encode_f(h(x_19), x_2) -> f(h(encArg(x_19)), a) [0] encode_f(cons_f(x_110, x_24), a) -> f(f(encArg(x_110), encArg(x_24)), a) [0] encode_f(cons_f(x_110, x_24), h(x_115)) -> f(f(encArg(x_110), encArg(x_24)), h(encArg(x_115))) [0] encode_f(cons_f(x_110, x_24), cons_f(x_116, x_27)) -> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) [0] encode_f(cons_f(x_110, x_24), x_2) -> f(f(encArg(x_110), encArg(x_24)), a) [0] encode_f(x_1, a) -> f(a, a) [0] encode_f(x_1, h(x_117)) -> f(a, h(encArg(x_117))) [0] encode_f(x_1, cons_f(x_118, x_28)) -> f(a, f(encArg(x_118), encArg(x_28))) [0] encode_f(x_1, x_2) -> f(a, a) [0] encode_a -> a [0] encode_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] encode_h(v0) -> a [0] f(v0, v1) -> a [0] The TRS has the following type information: f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f c_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f a :: a:c_f:h:cons_f h :: a:c_f:h:cons_f -> a:c_f:h:cons_f encArg :: a:c_f:h:cons_f -> a:c_f:h:cons_f cons_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_f :: a:c_f:h:cons_f -> a:c_f:h:cons_f -> a:c_f:h:cons_f encode_a :: a:c_f:h:cons_f encode_h :: a:c_f:h:cons_f -> a:c_f:h:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + x_1 + 0, x_1 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(0, 1 + encArg(x_11)) :|: x_11 >= 0, z = 1 + 0 + (1 + x_11) encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17) encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: z = 1 + (1 + x_1') + 0, x_1' >= 0 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_2 >= 0, z' = x_2, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(x_115)) :|: x_115 >= 0, z' = 1 + x_115, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: x_1 >= 0, z = x_1, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_2 >= 0, z' = x_2, z = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = x_1, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_111)) :|: z' = 1 + x_111, z = 0, x_111 >= 0 encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_117)) :|: x_1 >= 0, x_117 >= 0, z' = 1 + x_117, z = x_1 encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z = 1 + x_19, x_19 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 0) :|: z = 1 + x_19, z' = 0, x_19 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 0) :|: x_2 >= 0, z' = x_2, z = 1 + x_19, x_19 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 1 + encArg(x_113)) :|: x_113 >= 0, z = 1 + x_19, z' = 1 + x_113, x_19 >= 0 encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_h(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_h(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 f(z, z') -{ 2 }-> f(0, f(f(f(0, f(f(f(0, 0), 1 + 0), y')), 1 + 0), y)) :|: y >= 0, y' >= 0, z = 1 + 0 + y', z' = 1 + 0 + y f(z, z') -{ 1 }-> f(0, f(f(0, 1 + 0), y)) :|: x >= 0, y >= 0, z = x, z' = 1 + 0 + y f(z, z') -{ 1 }-> f(0, f(f(1 + 0 + x, 1 + 0), y)) :|: x >= 0, y >= 0, z = x, z' = 1 + 0 + y f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 ---------------------------------------- (31) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0 encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17) encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2 encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0 encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_a -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0 encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0 encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0 encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_h(z) -{ 0 }-> 0 :|: z >= 0 encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 f(z, z') -{ 2 }-> f(0, f(f(f(0, f(f(f(0, 0), 1 + 0), z - 1)), 1 + 0), z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 f(z, z') -{ 1 }-> f(0, f(f(0, 1 + 0), z' - 1)) :|: z >= 0, z' - 1 >= 0 f(z, z') -{ 1 }-> f(0, f(f(1 + 0 + z, 1 + 0), z' - 1)) :|: z >= 0, z' - 1 >= 0 f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 ---------------------------------------- (33) 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_h(v0) -> null_encode_h [0] f(v0, v1) -> null_f [0] And the following fresh constants: null_encArg, null_encode_f, null_encode_a, null_encode_h, null_f ---------------------------------------- (34) 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(a, y)) -> f(a, f(f(f(a, x), h(a)), y)) [1] encArg(a) -> a [0] encArg(h(x_1)) -> h(encArg(x_1)) [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_h(x_1) -> h(encArg(x_1)) [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_h(v0) -> null_encode_h [0] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f c_f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f a :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f h :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f encArg :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f cons_f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f encode_f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f encode_a :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f encode_h :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f -> a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f null_encArg :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f null_encode_f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f null_encode_a :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f null_encode_h :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f null_f :: a:c_f:h:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_h:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (35) 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 null_encArg => 0 null_encode_f => 0 null_encode_a => 0 null_encode_h => 0 null_f => 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_a -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_h(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_h(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 f(z, z') -{ 1 }-> f(0, f(f(f(0, x), 1 + 0), y)) :|: x >= 0, y >= 0, z = x, z' = 1 + 0 + y f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_h(z0) -> h(encArg(z0)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(a) -> c ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c3(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A -> c4 ENCODE_H(z0) -> c5(ENCARG(z0)) F(z0, z1) -> c6 F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(f(a, z0), h(a)), F(a, z0)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(f(a, z0), h(a)), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_h_1 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_A, ENCODE_H_1, F_2 Compound Symbols: c, c1_1, c2_3, c3_3, c4, c5_1, c6, c7_4 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_H(z0) -> c5(ENCARG(z0)) Removed 3 trailing nodes: F(z0, z1) -> c6 ENCODE_A -> c4 ENCARG(a) -> c ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_h(z0) -> h(encArg(z0)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c3(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(f(a, z0), h(a)), F(a, z0)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(f(a, z0), h(a)), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_h_1 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_3, c3_3, c7_4 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_h(z0) -> h(encArg(z0)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c3(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_h_1 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_3, c3_3, c7_3 ---------------------------------------- (43) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_h(z0) -> h(encArg(z0)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_h_1 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c7_3, c_1 ---------------------------------------- (45) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a encode_h(z0) -> h(encArg(z0)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a, encode_h_1 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c7_3, c_1 ---------------------------------------- (47) 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_h(z0) -> h(encArg(z0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c7_3, c_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_f(x0, a)) -> c2(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, h(z0))) -> c2(F(encArg(x0), h(encArg(z0))), ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, a)) -> c2(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, h(z0))) -> c2(F(encArg(x0), h(encArg(z0))), ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c7_3, c_1, c2_3 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) S tuples: F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c7_3, c_1, c2_3, c2_1, c2_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, c_f(a, z1)) -> c7(F(a, f(f(f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) by F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c_1, c2_3, c2_1, c2_2, c7_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) by ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ENCODE_F(x0, h(z0)) -> c(F(encArg(x0), h(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ENCODE_F(x0, h(z0)) -> c(F(encArg(x0), h(encArg(z0)))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(x0, h(z0)) -> c(F(encArg(x0), h(encArg(z0)))) ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, cons_f(z0, z1))) -> c2(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) by ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(h(z0), x1)) -> c2(F(h(encArg(z0)), encArg(x1)), ENCARG(h(z0)), ENCARG(x1)) by ENCARG(cons_f(h(x0), a)) -> c2(F(h(encArg(x0)), a), ENCARG(h(x0)), ENCARG(a)) ENCARG(cons_f(h(x0), h(z0))) -> c2(F(h(encArg(x0)), h(encArg(z0))), ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(h(x0), cons_f(z0, z1))) -> c2(F(h(encArg(x0)), f(encArg(z0), encArg(z1))), ENCARG(h(x0)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(x0), a)) -> c2(F(h(encArg(x0)), a), ENCARG(h(x0)), ENCARG(a)) ENCARG(cons_f(h(x0), h(z0))) -> c2(F(h(encArg(x0)), h(encArg(z0))), ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_3, c2_1, c2_2, c7_3, c_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_f(z0, z1), x1)) -> c2(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) by ENCARG(cons_f(cons_f(x0, x1), a)) -> c2(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(F(f(encArg(x0), encArg(x1)), h(encArg(z0))), ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c2(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, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c2(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(F(f(encArg(x0), encArg(x1)), h(encArg(z0))), ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(a, x1)) -> c2(F(a, encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(a, a)) -> c2(F(a, a), ENCARG(a)) ENCARG(cons_f(a, h(z0))) -> c2(F(a, h(encArg(z0))), ENCARG(h(z0))) ENCARG(cons_f(a, cons_f(z0, z1))) -> c2(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, a)) -> c2(F(a, a), ENCARG(a)) ENCARG(cons_f(a, h(z0))) -> c2(F(a, h(encArg(z0))), ENCARG(h(z0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCARG(cons_f(a, a)) -> c2(F(a, a), ENCARG(a)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(F(a, h(encArg(z0))), ENCARG(h(z0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) S tuples: F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) by F(x0, c_f(a, x1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(x0, c_f(a, x1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) S tuples: F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) S tuples: F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1, c7_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) by F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) S tuples: F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1, c7_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) by F(x0, c_f(a, z1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(x0, c_f(a, z1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(x0, c_f(a, z1)) -> c7(F(a, c_f(c_f(f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1, c7_2 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1, c7_2 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z1, c_f(a, x1)) -> c7(F(a, f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) by F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(c_f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, x0), h(a)), x1)), F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) S tuples: F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c7_3, c_1, c2_3, c7_1, c7_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) by F(c_f(a, x0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), x0)), h(a)), z1)), F(f(f(a, c_f(a, x0)), h(a)), z1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), x0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, x0)), h(a)), c_f(a, z1)), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(x0, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) by ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(a, x1) -> c(F(a, encArg(x1))) by ENCODE_F(a, a) -> c(F(a, a)) ENCODE_F(a, h(z0)) -> c(F(a, h(encArg(z0)))) ENCODE_F(a, cons_f(z0, z1)) -> c(F(a, f(encArg(z0), encArg(z1)))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(a, a) -> c(F(a, a)) ENCODE_F(a, h(z0)) -> c(F(a, h(encArg(z0)))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(a, h(z0)) -> c(F(a, h(encArg(z0)))) ENCODE_F(a, a) -> c(F(a, a)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(h(z0), x1) -> c(F(h(encArg(z0)), encArg(x1))) by ENCODE_F(h(x0), a) -> c(F(h(encArg(x0)), a)) ENCODE_F(h(x0), h(z0)) -> c(F(h(encArg(x0)), h(encArg(z0)))) ENCODE_F(h(x0), cons_f(z0, z1)) -> c(F(h(encArg(x0)), f(encArg(z0), encArg(z1)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(x0), a) -> c(F(h(encArg(x0)), a)) ENCODE_F(h(x0), h(z0)) -> c(F(h(encArg(x0)), h(encArg(z0)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(h(x0), h(z0)) -> c(F(h(encArg(x0)), h(encArg(z0)))) ENCODE_F(h(x0), a) -> c(F(h(encArg(x0)), a)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c2_1, c2_2, c_1, c2_3, c7_3, c7_1, c7_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) by ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ENCODE_F(cons_f(x0, x1), h(z0)) -> c(F(f(encArg(x0), encArg(x1)), h(encArg(z0)))) ENCODE_F(cons_f(x0, x1), cons_f(z0, z1)) -> c(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ENCODE_F(cons_f(x0, x1), h(z0)) -> c(F(f(encArg(x0), encArg(x1)), h(encArg(z0)))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCODE_F(cons_f(x0, x1), h(z0)) -> c(F(f(encArg(x0), encArg(x1)), h(encArg(z0)))) ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) by F(z0, c_f(a, z1)) -> c7(F(a, c_f(c_f(c_f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, c_f(a, z1)) -> c7(F(a, c_f(c_f(c_f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) S tuples: F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(z0, c_f(a, z1)) -> c7(F(a, c_f(c_f(c_f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) by F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(h(z0)) -> c1(ENCARG(z0)) ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_1, c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(h(z0)) -> c1(ENCARG(z0)) by ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(h(a)), ENCARG(x1)) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) by F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) by F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) by F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(z1, c_f(a, x1)) -> c7(F(a, c_f(f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(x0, c_f(a, z1)) -> c7(F(a, c_f(f(c_f(a, x0), h(a)), z1)), F(f(f(a, x0), h(a)), z1), F(a, x0)) by F(z0, c_f(a, z1)) -> c7(F(a, c_f(c_f(c_f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, z1)) -> c7(F(a, c_f(c_f(c_f(a, z0), h(a)), z1)), F(f(f(a, z0), h(a)), z1), F(a, z0)) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(h(z0)) -> h(encArg(z0)) encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, z1) -> c_f(z0, z1) f(z0, c_f(a, z1)) -> f(a, f(f(f(a, z0), h(a)), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c2(ENCARG(x0)) ENCARG(cons_f(x0, h(z0))) -> c2(ENCARG(x0), ENCARG(h(z0))) ENCARG(cons_f(x0, cons_f(x1, x2))) -> c2(F(encArg(x0), c_f(encArg(x1), encArg(x2))), ENCARG(x0), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c2(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, h(z0)))) -> c2(F(encArg(x0), f(encArg(x1), h(encArg(z0)))), ENCARG(x0), ENCARG(cons_f(x1, h(z0)))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c2(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1)))) ENCARG(cons_f(x0, cons_f(a, x2))) -> c2(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(h(z0), x2))) -> c2(F(encArg(x0), f(h(encArg(z0)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(h(z0), x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c2(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(h(z0), cons_f(x1, x2))) -> c2(F(h(encArg(z0)), f(encArg(x1), encArg(x2))), ENCARG(h(z0)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c2(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(a, cons_f(x1, x2))) -> c2(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(h(h(z0)), x1)) -> c2(F(h(h(encArg(z0))), encArg(x1)), ENCARG(h(h(z0))), ENCARG(x1)) ENCARG(cons_f(h(cons_f(z0, z1)), x1)) -> c2(F(h(f(encArg(z0), encArg(z1))), encArg(x1)), ENCARG(h(cons_f(z0, z1))), ENCARG(x1)) ENCARG(cons_f(h(x0), x1)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), a)) -> c2(ENCARG(h(x0))) ENCARG(cons_f(h(x0), h(z0))) -> c2(ENCARG(h(x0)), ENCARG(h(z0))) ENCARG(cons_f(cons_f(x0, x1), x2)) -> c2(F(c_f(encArg(x0), encArg(x1)), encArg(x2)), ENCARG(cons_f(x0, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c2(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, h(z0)), x2)) -> c2(F(f(encArg(x0), h(encArg(z0))), encArg(x2)), ENCARG(cons_f(x0, h(z0))), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c2(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)) -> c2(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(h(z0), x1), x2)) -> c2(F(f(h(encArg(z0)), encArg(x1)), encArg(x2)), ENCARG(cons_f(h(z0), x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c2(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)) -> c2(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(cons_f(x0, x1), h(z0))) -> c2(ENCARG(cons_f(x0, x1)), ENCARG(h(z0))) ENCARG(cons_f(a, h(z0))) -> c2(ENCARG(h(z0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, c_f(f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(z1, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(c_f(a, z1), h(a))), h(a)), x1))), F(f(f(a, z1), h(a)), c_f(a, x1)), F(a, z1)) F(c_f(a, z1), c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(f(f(a, a), h(a)), z1)), h(a))), h(a)), x1))), F(f(f(a, c_f(a, z1)), h(a)), c_f(a, x1)), F(a, c_f(a, z1))) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(c_f(f(a, f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) ENCODE_F(x0, cons_f(x1, x2)) -> c(F(encArg(x0), c_f(encArg(x1), encArg(x2)))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) ENCODE_F(x0, cons_f(x1, h(z0))) -> c(F(encArg(x0), f(encArg(x1), h(encArg(z0))))) ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1))))) ENCODE_F(x0, cons_f(a, x2)) -> c(F(encArg(x0), f(a, encArg(x2)))) ENCODE_F(x0, cons_f(h(z0), x2)) -> c(F(encArg(x0), f(h(encArg(z0)), encArg(x2)))) ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2)))) ENCODE_F(a, cons_f(x1, x2)) -> c(F(a, f(encArg(x1), encArg(x2)))) ENCODE_F(h(z0), cons_f(x1, x2)) -> c(F(h(encArg(z0)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(h(a), x1) -> c(F(h(a), encArg(x1))) ENCODE_F(h(h(z0)), x1) -> c(F(h(h(encArg(z0))), encArg(x1))) ENCODE_F(h(cons_f(z0, z1)), x1) -> c(F(h(f(encArg(z0), encArg(z1))), encArg(x1))) ENCODE_F(cons_f(x0, x1), x2) -> c(F(c_f(encArg(x0), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), encArg(x2))) ENCODE_F(cons_f(x0, h(z0)), x2) -> c(F(f(encArg(x0), h(encArg(z0))), encArg(x2))) ENCODE_F(cons_f(x0, cons_f(z0, z1)), x2) -> c(F(f(encArg(x0), f(encArg(z0), encArg(z1))), encArg(x2))) ENCODE_F(cons_f(a, x1), x2) -> c(F(f(a, encArg(x1)), encArg(x2))) ENCODE_F(cons_f(h(z0), x1), x2) -> c(F(f(h(encArg(z0)), encArg(x1)), encArg(x2))) ENCODE_F(cons_f(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) ENCARG(h(h(y0))) -> c1(ENCARG(h(y0))) ENCARG(h(cons_f(y0, a))) -> c1(ENCARG(cons_f(y0, a))) ENCARG(h(cons_f(y0, h(y1)))) -> c1(ENCARG(cons_f(y0, h(y1)))) ENCARG(h(cons_f(y0, cons_f(y1, y2)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, y2)))) ENCARG(h(cons_f(y0, cons_f(y1, a)))) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(h(cons_f(y0, cons_f(y1, h(y2))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, h(y2))))) ENCARG(h(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(h(cons_f(y0, cons_f(a, y1)))) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(h(cons_f(y0, cons_f(h(y1), y2)))) -> c1(ENCARG(cons_f(y0, cons_f(h(y1), y2)))) ENCARG(h(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(h(cons_f(h(y0), cons_f(y1, y2)))) -> c1(ENCARG(cons_f(h(y0), cons_f(y1, y2)))) ENCARG(h(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(h(cons_f(a, cons_f(y0, y1)))) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(h(cons_f(h(a), y0))) -> c1(ENCARG(cons_f(h(a), y0))) ENCARG(h(cons_f(h(h(y0)), y1))) -> c1(ENCARG(cons_f(h(h(y0)), y1))) ENCARG(h(cons_f(h(cons_f(y0, y1)), y2))) -> c1(ENCARG(cons_f(h(cons_f(y0, y1)), y2))) ENCARG(h(cons_f(h(y0), y1))) -> c1(ENCARG(cons_f(h(y0), y1))) ENCARG(h(cons_f(h(y0), a))) -> c1(ENCARG(cons_f(h(y0), a))) ENCARG(h(cons_f(h(y0), h(y1)))) -> c1(ENCARG(cons_f(h(y0), h(y1)))) ENCARG(h(cons_f(cons_f(y0, y1), y2))) -> c1(ENCARG(cons_f(cons_f(y0, y1), y2))) ENCARG(h(cons_f(cons_f(y0, a), y1))) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(h(cons_f(cons_f(y0, h(y1)), y2))) -> c1(ENCARG(cons_f(cons_f(y0, h(y1)), y2))) ENCARG(h(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(h(cons_f(cons_f(a, y0), y1))) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(h(cons_f(cons_f(h(y0), y1), y2))) -> c1(ENCARG(cons_f(cons_f(h(y0), y1), y2))) ENCARG(h(cons_f(cons_f(cons_f(y0, y1), y2), y3))) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(h(cons_f(cons_f(y0, y1), a))) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(h(cons_f(cons_f(y0, y1), h(y2)))) -> c1(ENCARG(cons_f(cons_f(y0, y1), h(y2)))) ENCARG(h(cons_f(a, h(y0)))) -> c1(ENCARG(cons_f(a, h(y0)))) ENCARG(cons_f(h(a), x1)) -> c2(F(h(a), encArg(x1)), ENCARG(x1)) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) S tuples: F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1)) F(x0, c_f(a, x1)) -> c7(F(f(f(a, x0), h(a)), x1), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, c_f(a, f(f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, c_f(f(f(a, f(f(a, x0), h(a))), h(a)), z1))), F(f(f(a, x0), h(a)), c_f(a, z1)), F(a, x0)) F(x0, c_f(a, c_f(a, c_f(a, z1)))) -> c7(F(a, f(a, f(a, f(f(f(a, f(f(a, f(f(a, x0), h(a))), h(a))), h(a)), z1)))), F(f(f(a, x0), h(a)), c_f(a, c_f(a, z1))), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(c_f(f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(c_f(a, f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(a, f(a, f(f(f(a, c_f(f(a, x0), h(a))), h(a)), x1))), F(f(f(a, x0), h(a)), c_f(a, x1)), F(a, x0)) F(x0, c_f(a, c_f(a, x1))) -> c7(F(f(f(a, x0), h(a)), c_f(a, x1))) F(z1, c_f(a, x1)) -> c7(F(a, f(c_f(c_f(a, z1), h(a)), x1)), F(f(f(a, z1), h(a)), x1), F(a, z1)) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(c_f(a, f(f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z1), c_f(a, x1)) -> c7(F(a, f(f(f(a, c_f(f(f(a, a), h(a)), z1)), h(a)), x1)), F(f(f(a, c_f(a, z1)), h(a)), x1), F(a, c_f(a, z1))) F(c_f(a, c_f(a, z1)), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(a, f(f(f(a, f(f(a, a), h(a))), h(a)), z1))), h(a)), x1)), F(f(f(a, c_f(a, c_f(a, z1))), h(a)), x1), F(a, c_f(a, c_f(a, z1)))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(c_f(f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(a, f(f(f(a, f(f(c_f(a, a), h(a)), x0)), h(a)), x1)), F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, x0), c_f(a, x1)) -> c7(F(f(f(a, c_f(a, x0)), h(a)), x1), F(a, c_f(a, x0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, c_f(f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) F(z0, c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, c_f(c_f(a, z0), h(a))), h(a)), z1))), F(f(f(a, z0), h(a)), c_f(a, z1)), F(a, z0)) F(c_f(a, z0), c_f(a, c_f(a, z1))) -> c7(F(a, f(a, f(f(f(a, f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a))), h(a)), z1))), F(f(f(a, c_f(a, z0)), h(a)), c_f(a, z1)), F(a, c_f(a, z0))) F(c_f(a, z0), c_f(a, z1)) -> c7(F(a, f(c_f(f(a, f(c_f(f(a, a), h(a)), z0)), h(a)), z1)), F(f(f(a, c_f(a, z0)), h(a)), z1), F(a, c_f(a, z0))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c2_1, c2_2, c2_3, c7_3, c7_1, c7_2, c_1, c1_1