/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 177 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) typed CpxTrs (11) OrderProof [LOWER BOUND(ID), 0 ms] (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 619 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxWeightedTrs (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxTypedWeightedTrs (23) CompletionProof [UPPER BOUND(ID), 0 ms] (24) CpxTypedWeightedCompleteTrs (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (44) CdtProblem (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 109 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 67 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 11 ms] (90) CdtProblem (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 17 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 24 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 5 ms] (96) CdtProblem (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 12 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 15 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 10 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 45 ms] (114) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(a) -> a encArg(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 ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a 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 Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a 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 Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a 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 Rewrite Strategy: INNERMOST ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) 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, f(y, a)) -> f(a, f(f(x, a), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a 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 Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) encArg(a) -> a 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 Types: f :: a:cons_f -> a:cons_f -> a:cons_f a :: a:cons_f encArg :: a:cons_f -> a:cons_f cons_f :: a:cons_f -> a:cons_f -> a:cons_f encode_f :: a:cons_f -> a:cons_f -> a:cons_f encode_a :: a:cons_f hole_a:cons_f1_0 :: a:cons_f gen_a:cons_f2_0 :: Nat -> a:cons_f ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Innermost TRS: Rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) encArg(a) -> a 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 Types: f :: a:cons_f -> a:cons_f -> a:cons_f a :: a:cons_f encArg :: a:cons_f -> a:cons_f cons_f :: a:cons_f -> a:cons_f -> a:cons_f encode_f :: a:cons_f -> a:cons_f -> a:cons_f encode_a :: a:cons_f hole_a:cons_f1_0 :: a:cons_f gen_a:cons_f2_0 :: Nat -> a:cons_f Generator Equations: gen_a:cons_f2_0(0) <=> a gen_a:cons_f2_0(+(x, 1)) <=> cons_f(a, gen_a: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 ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_a:cons_f2_0(n9_0)) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_a:cons_f2_0(0)) Induction Step: encArg(gen_a:cons_f2_0(+(n9_0, 1))) ->_R^Omega(0) f(encArg(a), encArg(gen_a:cons_f2_0(n9_0))) ->_R^Omega(0) f(a, encArg(gen_a:cons_f2_0(n9_0))) ->_IH f(a, *3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (14) BOUNDS(1, INF) ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, f(y, a)) -> f(a, f(f(x, a), y)) encArg(a) -> a 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 S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, c_f(y, a)) -> f(a, f(f(x, a), y)) The (relative) TRS S consists of the following rules: encArg(a) -> a 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 f(x0, x1) -> c_f(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (19) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: f(x, c_f(y, a)) -> f(a, f(f(x, a), y)) [1] encArg(a) -> a [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] f(x0, x1) -> c_f(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, c_f(y, a)) -> f(a, f(f(x, a), y)) [1] encArg(a) -> a [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] f(x0, x1) -> c_f(x0, x1) [0] The TRS has the following type information: f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f c_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f a :: a:c_f:cons_f encArg :: a:c_f:cons_f -> a:c_f:cons_f cons_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_a :: a:c_f:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (23) 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 f_2 Due to the following rules being added: encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] f(v0, v1) -> a [0] And the following fresh constants: none ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, c_f(y, a)) -> f(a, f(f(x, a), y)) [1] encArg(a) -> a [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] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] f(v0, v1) -> a [0] The TRS has the following type information: f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f c_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f a :: a:c_f:cons_f encArg :: a:c_f:cons_f -> a:c_f:cons_f cons_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_a :: a:c_f:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (25) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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(y, a)) -> f(a, f(c_f(x, a), y)) [1] f(x, c_f(y, a)) -> f(a, f(a, y)) [1] encArg(a) -> a [0] encArg(cons_f(a, a)) -> f(a, a) [0] encArg(cons_f(a, cons_f(x_1'', x_2''))) -> f(a, f(encArg(x_1''), encArg(x_2''))) [0] encArg(cons_f(a, x_2)) -> f(a, 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'), cons_f(x_11, x_21))) -> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_11), encArg(x_21))) [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, cons_f(x_12, x_22))) -> f(a, f(encArg(x_12), encArg(x_22))) [0] encArg(cons_f(x_1, x_2)) -> f(a, a) [0] encode_f(a, a) -> f(a, a) [0] encode_f(a, cons_f(x_14, x_24)) -> f(a, f(encArg(x_14), encArg(x_24))) [0] encode_f(a, x_2) -> f(a, a) [0] encode_f(cons_f(x_13, x_23), a) -> f(f(encArg(x_13), encArg(x_23)), a) [0] encode_f(cons_f(x_13, x_23), cons_f(x_15, x_25)) -> f(f(encArg(x_13), encArg(x_23)), f(encArg(x_15), encArg(x_25))) [0] encode_f(cons_f(x_13, x_23), x_2) -> f(f(encArg(x_13), encArg(x_23)), a) [0] encode_f(x_1, a) -> f(a, a) [0] encode_f(x_1, cons_f(x_16, x_26)) -> f(a, f(encArg(x_16), encArg(x_26))) [0] encode_f(x_1, x_2) -> f(a, a) [0] encode_a -> a [0] f(x0, x1) -> c_f(x0, x1) [0] encArg(v0) -> a [0] encode_f(v0, v1) -> a [0] encode_a -> a [0] f(v0, v1) -> a [0] The TRS has the following type information: f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f c_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f a :: a:c_f:cons_f encArg :: a:c_f:cons_f -> a:c_f:cons_f cons_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_f :: a:c_f:cons_f -> a:c_f:cons_f -> a:c_f:cons_f encode_a :: a:c_f:cons_f Rewrite Strategy: INNERMOST ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21) encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0 encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_1''), encArg(x_2''))) :|: x_1'' >= 0, z = 1 + 0 + (1 + x_1'' + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_22))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_12 + x_22), x_12 >= 0, x_22 >= 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 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), f(encArg(x_15), encArg(x_25))) :|: x_15 >= 0, z' = 1 + x_15 + x_25, x_25 >= 0, z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), 0) :|: z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), 0) :|: z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0, x_2 >= 0, z' = x_2 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_14), encArg(x_24))) :|: x_14 >= 0, x_24 >= 0, z = 0, z' = 1 + x_14 + x_24 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, z' = 1 + x_16 + x_26, x_26 >= 0, z = x_1 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 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 f(z, z') -{ 1 }-> f(0, f(0, y)) :|: x >= 0, y >= 0, z' = 1 + y + 0, z = x f(z, z') -{ 1 }-> f(0, f(1 + x + 0, y)) :|: x >= 0, y >= 0, z' = 1 + y + 0, z = x 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 ---------------------------------------- (29) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21) encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0 encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_1''), encArg(x_2''))) :|: x_1'' >= 0, z = 1 + 0 + (1 + x_1'' + x_2''), x_2'' >= 0 encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_22))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_12 + x_22), x_12 >= 0, x_22 >= 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 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_a -{ 0 }-> 0 :|: encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), f(encArg(x_15), encArg(x_25))) :|: x_15 >= 0, z' = 1 + x_15 + x_25, x_25 >= 0, z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), 0) :|: z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0, z' = 0 encode_f(z, z') -{ 0 }-> f(f(encArg(x_13), encArg(x_23)), 0) :|: z = 1 + x_13 + x_23, x_13 >= 0, x_23 >= 0, z' >= 0 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_14), encArg(x_24))) :|: x_14 >= 0, x_24 >= 0, z = 0, z' = 1 + x_14 + x_24 encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: z >= 0, x_16 >= 0, z' = 1 + x_16 + x_26, x_26 >= 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 }-> 0 :|: z >= 0, z' >= 0 f(z, z') -{ 1 }-> f(0, f(0, z' - 1)) :|: z >= 0, z' - 1 >= 0 f(z, z') -{ 1 }-> f(0, f(1 + z + 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 ---------------------------------------- (31) 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] f(v0, v1) -> null_f [0] And the following fresh constants: null_encArg, null_encode_f, null_encode_a, null_f ---------------------------------------- (32) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(x, c_f(y, a)) -> f(a, f(f(x, a), y)) [1] encArg(a) -> a [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] 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] f(v0, v1) -> null_f [0] The TRS has the following type information: f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f c_f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f a :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f encArg :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f cons_f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f encode_f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f -> a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f encode_a :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f null_encArg :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f null_encode_f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f null_encode_a :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f null_f :: a:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_f Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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_f => 0 ---------------------------------------- (34) 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 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 f(z, z') -{ 1 }-> f(0, f(f(x, 0), y)) :|: x >= 0, y >= 0, z' = 1 + y + 0, z = x 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. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(a) -> c ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A -> c3 F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1), F(z0, a)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1), F(z0, a)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_A, F_2 Compound Symbols: c, c1_3, c2_3, c3, c4_3 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: ENCARG(a) -> c ENCODE_A -> c3 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1), F(z0, a)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1), F(z0, a)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c2_3, c4_3 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_F(z0, z1) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c2_3, c4_2 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) 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, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c4_2, c_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_F(z0, z1) -> c(ENCARG(z0)) ENCODE_F(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_a -> a f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: f_2, encArg_1, encode_f_2, encode_a Defined Pair Symbols: ENCARG_1, F_2, ENCODE_F_2 Compound Symbols: c1_3, c4_2, c_1 ---------------------------------------- (45) 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 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, 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_3, c4_2, c_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, a)) -> c1(F(encArg(x0), a), ENCARG(x0), ENCARG(a)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(a), ENCARG(x1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_2, c_1, c1_3 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCODE_F(z0, z1) -> c(F(encArg(z0), encArg(z1))) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCODE_F_2, ENCARG_1 Compound Symbols: c4_2, c_1, c1_3, c1_1, c1_2 ---------------------------------------- (51) 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, cons_f(z0, z1)) -> c(F(encArg(x0), f(encArg(z0), encArg(z1)))) ENCODE_F(a, x1) -> c(F(a, encArg(x1))) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) ENCODE_F(x0, a) -> c(F(encArg(x0), a)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_1, c1_2, c_1 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(x0, a) -> c(F(encArg(x0), a)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_1, c1_2, c_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(x0, cons_f(z0, z1))) -> c1(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) by ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(a), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_1, c1_2, c_1 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_3, c1_1, c1_2, c_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(cons_f(z0, z1), x1)) -> c1(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1)) by ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, x1), cons_f(z0, z1))) -> c1(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1))), ENCARG(cons_f(x0, x1)), ENCARG(cons_f(z0, z1))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(F(f(encArg(x0), encArg(x1)), a), ENCARG(cons_f(x0, x1)), ENCARG(a)) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c1_2, c_1, c1_3 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c1_2, c_1, c1_3 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_f(a, x1)) -> c1(F(a, encArg(x1)), ENCARG(x1)) by ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) ENCARG(cons_f(a, cons_f(z0, z1))) -> c1(F(a, f(encArg(z0), encArg(z1))), ENCARG(cons_f(z0, z1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCARG(cons_f(a, a)) -> c1(F(a, a), ENCARG(a)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) 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(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (67) 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, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCODE_F(a, x1) -> c(F(a, 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, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (69) 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, cons_f(z0, z1)) -> c(F(a, f(encArg(z0), encArg(z1)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(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)) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(a, a) -> c(F(a, a)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCODE_F(cons_f(z0, z1), x1) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1))) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c_1, c1_3, c1_2 ---------------------------------------- (73) 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), cons_f(z0, z1)) -> c(F(f(encArg(x0), encArg(x1)), f(encArg(z0), encArg(z1)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c1_3, c1_2, c_1 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_F(cons_f(x0, x1), a) -> c(F(f(encArg(x0), encArg(x1)), a)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) S tuples: F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: F_2, ENCARG_1, ENCODE_F_2 Compound Symbols: c4_2, c1_1, c1_3, c1_2, c_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(z0, f(z1, a)) -> c4(F(a, f(f(z0, a), z1)), F(f(z0, a), z1)) by F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a)) F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a)) F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a)), F(f(z0, a), a)) F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c1_3, c1_2, c_1, c4_2 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) S tuples: F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c1_3, c1_2, c_1, c4_2, c4_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(z0, f(f(y1, a), a)) -> c4(F(a, f(f(z0, a), f(y1, a))), F(f(z0, a), f(y1, a))) by F(x0, f(f(z1, a), a)) -> c4(F(a, f(a, f(f(f(x0, a), a), z1))), F(f(x0, a), f(z1, a))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(z1, a), a)) -> c4(F(a, f(a, f(f(f(x0, a), a), z1))), F(f(x0, a), f(z1, a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(z1, a), a)) -> c4(F(a, f(a, f(f(f(x0, a), a), z1))), F(f(x0, a), f(z1, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c1_3, c1_2, c_1, c4_1, c4_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(x0, f(f(z1, a), a)) -> c4(F(a, f(a, f(f(f(x0, a), a), z1))), F(f(x0, a), f(z1, a))) by F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_1, c1_3, c1_2, c_1, c4_1, c4_2 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(x0, a)) -> c1(ENCARG(x0)) by ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c1_1, c_1, c4_1, c4_2 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(x0, f(f(a, a), a)) -> c4(F(a, f(a, f(f(a, a), f(f(x0, a), a)))), F(f(x0, a), f(a, a))) by F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(f(f(a, a), a), f(z0, a))))), F(f(z0, a), f(a, a))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(f(f(a, a), a), f(z0, a))))), F(f(z0, a), f(a, a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(f(f(a, a), a), f(z0, a))))), F(f(z0, a), f(a, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c1_1, c_1, c4_1, c4_2 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(f(f(a, a), a), f(z0, a))))), F(f(z0, a), f(a, a))) by F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(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)) -> c1(ENCARG(cons_f(x0, x1))) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c1_1, c_1, c4_1, c4_2 ---------------------------------------- (91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(cons_f(x0, x1), a)) -> c1(ENCARG(cons_f(x0, x1))) by ENCARG(cons_f(cons_f(z0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(z0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(z0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(z0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(z0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(z0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(z0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), z1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), z1), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), z1))) ENCARG(cons_f(cons_f(cons_f(a, y0), z1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), z1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), z1), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), z1))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c4_1, c4_2, c1_1 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, f(f(x1, a), a)) -> c4(F(f(x0, a), f(x1, a))) by F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) S tuples: F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c4_1, c4_2, c1_1 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(z0, f(a, a)) -> c4(F(a, f(f(z0, a), a))) by F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) S tuples: F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c4_2, c1_1, c4_1 ---------------------------------------- (97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace F(x0, f(f(f(z1, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(x0, a), a), a), z1)))), F(f(x0, a), f(f(z1, a), a))) by F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(y1, a))))), F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(y1, a))))), F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(y1, a))))), F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(y1, a))))), F(f(z0, a), f(f(f(y1, a), a), a))) by F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), a)))), F(f(z0, a), f(f(a, a), a))) by F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(f(f(y1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(f(y1, a), a))))), F(f(z0, a), f(f(f(f(y1, a), a), a), a))) by F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(f(f(f(z0, a), a), a), f(a, a))))), F(f(z0, a), f(f(f(a, a), a), a))) by F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(f(a, a), f(f(f(z0, a), a), a))))), F(f(z0, a), f(f(a, a), a))) by F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), f(z1, a)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) by F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), a))))), F(f(z0, a), f(f(f(a, a), a), a))) by F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(a, a), f(f(f(f(z0, a), a), a), a)))))), F(f(z0, a), f(f(f(a, a), a), a))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(a, a), f(f(f(f(z0, a), a), a), a)))))), F(f(z0, a), f(f(f(a, a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(a, a), f(f(f(f(z0, a), a), a), a)))))), F(f(z0, a), f(f(f(a, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(cons_f(cons_f(y0, a), a)) -> c1(ENCARG(cons_f(y0, a))) by ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), a))) ENCARG(cons_f(cons_f(cons_f(a, y0), a), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a) -> a encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) f(z0, f(z1, a)) -> f(a, f(f(z0, a), z1)) Tuples: ENCARG(cons_f(x0, cons_f(x1, a))) -> c1(F(encArg(x0), f(encArg(x1), a)), ENCARG(x0), ENCARG(cons_f(x1, a))) ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c1(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))) -> c1(F(encArg(x0), f(a, encArg(x2))), ENCARG(x0), ENCARG(cons_f(a, x2))) ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c1(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2))) ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c1(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))) -> c1(F(a, f(encArg(x1), encArg(x2))), ENCARG(cons_f(x1, x2))) ENCARG(cons_f(cons_f(x0, a), x2)) -> c1(F(f(encArg(x0), a), encArg(x2)), ENCARG(cons_f(x0, a)), ENCARG(x2)) ENCARG(cons_f(cons_f(x0, cons_f(z0, z1)), x2)) -> c1(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)) -> c1(F(f(a, encArg(x1)), encArg(x2)), ENCARG(cons_f(a, x1)), ENCARG(x2)) ENCARG(cons_f(cons_f(cons_f(z0, z1), x1), x2)) -> c1(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2)), ENCARG(cons_f(cons_f(z0, z1), x1)), ENCARG(x2)) ENCODE_F(x0, cons_f(x1, a)) -> c(F(encArg(x0), f(encArg(x1), a))) 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(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(cons_f(z0, z1), cons_f(x1, x2)) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2)))) ENCODE_F(cons_f(x0, a), x2) -> c(F(f(encArg(x0), a), 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(cons_f(z0, z1), x1), x2) -> c(F(f(f(encArg(z0), encArg(z1)), encArg(x1)), encArg(x2))) ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, a)))) ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a)) -> c1(ENCARG(cons_f(y0, cons_f(y1, cons_f(y2, y3))))) ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a)) -> c1(ENCARG(cons_f(y0, cons_f(a, y1)))) ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a)) -> c1(ENCARG(cons_f(y0, cons_f(cons_f(y1, y2), y3)))) ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), cons_f(y2, y3)))) ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a)) -> c1(ENCARG(cons_f(a, cons_f(y0, y1)))) ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), y1))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), y3))) ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), y1))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), y3))) ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, y1), a))) F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) ENCARG(cons_f(cons_f(cons_f(y0, a), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, a)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(a, y1)), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a))) ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a)) -> c1(ENCARG(cons_f(cons_f(a, cons_f(y0, y1)), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, y0), y1), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), a), a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(a, a), f(f(f(f(z0, a), a), a), a)))))), F(f(z0, a), f(f(f(a, a), a), a))) ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), a), a)) -> c1(ENCARG(cons_f(cons_f(y0, cons_f(y1, y2)), a))) ENCARG(cons_f(cons_f(cons_f(a, y0), a), a)) -> c1(ENCARG(cons_f(cons_f(a, y0), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, y1), y2), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, a), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, a)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(y1, cons_f(y2, y3))), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(a, y1)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(y0, cons_f(cons_f(y1, y2), y3)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), cons_f(y2, y3)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(a, cons_f(y0, y1)), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, a), y1), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, cons_f(y1, y2)), y3), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(a, y0), y1), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), y2), y3), a), a))) ENCARG(cons_f(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a), a)) -> c1(ENCARG(cons_f(cons_f(cons_f(cons_f(y0, y1), a), a), a))) S tuples: F(z0, f(f(a, a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(a, a), a), a), z0))))), F(f(z0, a), f(a, a))) F(z0, f(f(a, a), a)) -> c4(F(f(z0, a), f(a, a))) F(z0, f(f(f(f(y1, a), a), a), a)) -> c4(F(f(z0, a), f(f(f(y1, a), a), a))) F(z0, f(f(f(y1, a), a), a)) -> c4(F(f(z0, a), f(f(y1, a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(f(z0, a), f(f(a, a), a))) F(f(y1, a), f(a, a)) -> c4(F(a, f(f(f(y1, a), a), a))) F(a, f(a, a)) -> c4(F(a, f(f(a, a), a))) F(f(f(y1, a), a), f(a, a)) -> c4(F(a, f(f(f(f(y1, a), a), a), a))) F(f(a, a), f(a, a)) -> c4(F(a, f(f(f(a, a), a), a))) F(z0, f(f(f(f(z1, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(f(f(z0, a), a), a), a), z1))))), F(f(z0, a), f(f(f(z1, a), a), a))) F(z0, f(f(f(a, a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(f(a, a), a), f(f(z0, a), a)))))), F(f(z0, a), f(f(a, a), a))) F(z0, f(f(f(f(f(z1, a), a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(a, f(f(f(f(f(f(z0, a), a), a), a), a), z1)))))), F(f(z0, a), f(f(f(f(z1, a), a), a), a))) F(z0, f(f(f(f(a, a), a), a), a)) -> c4(F(a, f(a, f(a, f(a, f(f(a, a), f(f(f(f(z0, a), a), a), a)))))), F(f(z0, a), f(f(f(a, a), a), a))) K tuples:none Defined Rule Symbols: encArg_1, f_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, F_2 Compound Symbols: c1_3, c1_2, c_1, c1_1, c4_2, c4_1