/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 58 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), 503 ms] (14) BOUNDS(1, INF) (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRelTRS (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxWeightedTrs (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTypedWeightedTrs (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (8) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: TRS: Rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c c :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c encArg :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c encode_c :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c hole_a:c:encArg:encode_a:encode_c1_0 :: a:c:encArg:encode_a:encode_c hole_cons_a:cons_c2_0 :: cons_a:cons_c gen_cons_a:cons_c3_0 :: Nat -> cons_a:cons_c ---------------------------------------- (11) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, c, encArg They will be analysed ascendingly in the following order: a = c a < encArg c < encArg ---------------------------------------- (12) Obligation: TRS: Rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c c :: a:c:encArg:encode_a:encode_c -> a:c:encArg:encode_a:encode_c encArg :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c encode_c :: cons_a:cons_c -> a:c:encArg:encode_a:encode_c hole_a:c:encArg:encode_a:encode_c1_0 :: a:c:encArg:encode_a:encode_c hole_cons_a:cons_c2_0 :: cons_a:cons_c gen_cons_a:cons_c3_0 :: Nat -> cons_a:cons_c Generator Equations: gen_cons_a:cons_c3_0(0) <=> hole_cons_a:cons_c2_0 gen_cons_a:cons_c3_0(+(x, 1)) <=> cons_a(gen_cons_a:cons_c3_0(x)) The following defined symbols remain to be analysed: c, a, encArg They will be analysed ascendingly in the following order: a = c a < encArg c < encArg ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_cons_a:cons_c3_0(+(1, n13_0))) -> *4_0, rt in Omega(0) Induction Base: encArg(gen_cons_a:cons_c3_0(+(1, 0))) Induction Step: encArg(gen_cons_a:cons_c3_0(+(1, +(n13_0, 1)))) ->_R^Omega(0) a(encArg(gen_cons_a:cons_c3_0(+(1, n13_0)))) ->_IH a(*4_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 (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(a(a(x1))) -> a(c(a(a(x1)))) c(c(c(x1))) -> a(x1) a(x1) -> x1 encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (17) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (18) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(x1) -> x1 a(c_a(c_a(x1))) -> a(c(a(a(x1)))) c(c_c(c_c(x1))) -> a(x1) The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) a(x0) -> c_a(x0) c(x0) -> c_c(x0) Rewrite Strategy: FULL ---------------------------------------- (19) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(x1) -> x1 a(c_a(c_a(x1))) -> a(c(a(a(x1)))) c(c_c(c_c(x1))) -> a(x1) The (relative) TRS S consists of the following rules: encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) a(x0) -> c_a(x0) c(x0) -> c_c(x0) Rewrite Strategy: INNERMOST ---------------------------------------- (21) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (22) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a(x1) -> x1 [1] a(c_a(c_a(x1))) -> a(c(a(a(x1)))) [1] c(c_c(c_c(x1))) -> a(x1) [1] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] c(x0) -> c_c(x0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (23) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(x1) -> x1 [1] a(c_a(c_a(x1))) -> a(c(a(a(x1)))) [1] c(c_c(c_c(x1))) -> a(x1) [1] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] c(x0) -> c_c(x0) [0] The TRS has the following type information: a :: c_a:c_c -> c_a:c_c c_a :: c_a:c_c -> c_a:c_c c :: c_a:c_c -> c_a:c_c c_c :: c_a:c_c -> c_a:c_c encArg :: cons_a:cons_c -> c_a:c_c cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> c_a:c_c encode_c :: cons_a:cons_c -> c_a:c_c Rewrite Strategy: INNERMOST ---------------------------------------- (25) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: encArg_1 encode_a_1 encode_c_1 a_1 c_1 Due to the following rules being added: encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_c(v0) -> const [0] a(v0) -> const [0] c(v0) -> const [0] And the following fresh constants: const, const1 ---------------------------------------- (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: a(x1) -> x1 [1] a(c_a(c_a(x1))) -> a(c(a(a(x1)))) [1] c(c_c(c_c(x1))) -> a(x1) [1] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_c(v0) -> const [0] a(v0) -> const [0] c(v0) -> const [0] The TRS has the following type information: a :: c_a:c_c:const -> c_a:c_c:const c_a :: c_a:c_c:const -> c_a:c_c:const c :: c_a:c_c:const -> c_a:c_c:const c_c :: c_a:c_c:const -> c_a:c_c:const encArg :: cons_a:cons_c -> c_a:c_c:const cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> c_a:c_c:const encode_c :: cons_a:cons_c -> c_a:c_c:const const :: c_a:c_c:const const1 :: cons_a:cons_c Rewrite Strategy: INNERMOST ---------------------------------------- (27) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (28) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(x1) -> x1 [1] a(c_a(c_a(x1))) -> a(c(a(x1))) [2] a(c_a(c_a(c_a(c_a(x1'))))) -> a(c(a(a(c(a(a(x1'))))))) [2] a(c_a(c_a(x1))) -> a(c(a(c_a(x1)))) [1] a(c_a(c_a(x1))) -> a(c(a(const))) [1] c(c_c(c_c(x1))) -> a(x1) [1] encArg(cons_a(cons_a(x_1'))) -> a(a(encArg(x_1'))) [0] encArg(cons_a(cons_c(x_1''))) -> a(c(encArg(x_1''))) [0] encArg(cons_a(x_1)) -> a(const) [0] encArg(cons_c(cons_a(x_11))) -> c(a(encArg(x_11))) [0] encArg(cons_c(cons_c(x_12))) -> c(c(encArg(x_12))) [0] encArg(cons_c(x_1)) -> c(const) [0] encode_a(cons_a(x_13)) -> a(a(encArg(x_13))) [0] encode_a(cons_c(x_14)) -> a(c(encArg(x_14))) [0] encode_a(x_1) -> a(const) [0] encode_c(cons_a(x_15)) -> c(a(encArg(x_15))) [0] encode_c(cons_c(x_16)) -> c(c(encArg(x_16))) [0] encode_c(x_1) -> c(const) [0] a(x0) -> c_a(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> const [0] encode_a(v0) -> const [0] encode_c(v0) -> const [0] a(v0) -> const [0] c(v0) -> const [0] The TRS has the following type information: a :: c_a:c_c:const -> c_a:c_c:const c_a :: c_a:c_c:const -> c_a:c_c:const c :: c_a:c_c:const -> c_a:c_c:const c_c :: c_a:c_c:const -> c_a:c_c:const encArg :: cons_a:cons_c -> c_a:c_c:const cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> c_a:c_c:const encode_c :: cons_a:cons_c -> c_a:c_c:const const :: c_a:c_c:const const1 :: cons_a:cons_c Rewrite Strategy: INNERMOST ---------------------------------------- (29) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: const => 0 const1 => 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> x1 :|: x1 >= 0, z = x1 a(z) -{ 2 }-> a(c(a(x1))) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 2 }-> a(c(a(a(c(a(a(x1'))))))) :|: x1' >= 0, z = 1 + (1 + (1 + (1 + x1'))) a(z) -{ 1 }-> a(c(a(0))) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 1 }-> a(c(a(1 + x1))) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 1 }-> a(x1) :|: z = 1 + (1 + x1), x1 >= 0 c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> c(c(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0 encArg(z) -{ 0 }-> c(a(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11) encArg(z) -{ 0 }-> c(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(c(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> a(a(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0 encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> a(c(encArg(x_14))) :|: x_14 >= 0, z = 1 + x_14 encode_a(z) -{ 0 }-> a(a(encArg(x_13))) :|: z = 1 + x_13, x_13 >= 0 encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> c(c(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0 encode_c(z) -{ 0 }-> c(a(encArg(x_15))) :|: x_15 >= 0, z = 1 + x_15 encode_c(z) -{ 0 }-> c(0) :|: x_1 >= 0, z = x_1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (31) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> z :|: z >= 0 a(z) -{ 2 }-> a(c(a(a(c(a(a(z - 4))))))) :|: z - 4 >= 0 a(z) -{ 1 }-> a(c(a(0))) :|: z - 2 >= 0 a(z) -{ 2 }-> a(c(a(z - 2))) :|: z - 2 >= 0 a(z) -{ 1 }-> a(c(a(1 + (z - 2)))) :|: z - 2 >= 0 a(z) -{ 0 }-> 0 :|: z >= 0 a(z) -{ 0 }-> 1 + z :|: z >= 0 c(z) -{ 1 }-> a(z - 2) :|: z - 2 >= 0 c(z) -{ 0 }-> 0 :|: z >= 0 c(z) -{ 0 }-> 1 + z :|: z >= 0 encArg(z) -{ 0 }-> c(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> a(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encode_a(z) -{ 0 }-> a(c(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0 encode_a(z) -{ 0 }-> a(0) :|: z >= 0 encode_a(z) -{ 0 }-> 0 :|: z >= 0 encode_c(z) -{ 0 }-> c(c(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(a(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(0) :|: z >= 0 encode_c(z) -{ 0 }-> 0 :|: z >= 0 ---------------------------------------- (33) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: encArg(v0) -> null_encArg [0] encode_a(v0) -> null_encode_a [0] encode_c(v0) -> null_encode_c [0] a(v0) -> null_a [0] c(v0) -> null_c [0] And the following fresh constants: null_encArg, null_encode_a, null_encode_c, null_a, null_c, const ---------------------------------------- (34) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a(x1) -> x1 [1] a(c_a(c_a(x1))) -> a(c(a(a(x1)))) [1] c(c_c(c_c(x1))) -> a(x1) [1] encArg(cons_a(x_1)) -> a(encArg(x_1)) [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_a(x_1) -> a(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] a(x0) -> c_a(x0) [0] c(x0) -> c_c(x0) [0] encArg(v0) -> null_encArg [0] encode_a(v0) -> null_encode_a [0] encode_c(v0) -> null_encode_c [0] a(v0) -> null_a [0] c(v0) -> null_c [0] The TRS has the following type information: a :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c c_a :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c c :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c c_c :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c encArg :: cons_a:cons_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c cons_a :: cons_a:cons_c -> cons_a:cons_c cons_c :: cons_a:cons_c -> cons_a:cons_c encode_a :: cons_a:cons_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c encode_c :: cons_a:cons_c -> c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c null_encArg :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c null_encode_a :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c null_encode_c :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c null_a :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c null_c :: c_a:c_c:null_encArg:null_encode_a:null_encode_c:null_a:null_c const :: cons_a:cons_c Rewrite Strategy: INNERMOST ---------------------------------------- (35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_encArg => 0 null_encode_a => 0 null_encode_c => 0 null_a => 0 null_c => 0 const => 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: a(z) -{ 1 }-> x1 :|: x1 >= 0, z = x1 a(z) -{ 1 }-> a(c(a(a(x1)))) :|: z = 1 + (1 + x1), x1 >= 0 a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 1 }-> a(x1) :|: z = 1 + (1 + x1), x1 >= 0 c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 encArg(z) -{ 0 }-> c(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_a(z) -{ 0 }-> a(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> c(encArg(x_1)) :|: x_1 >= 0, z = x_1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0)) A(z0) -> c5 A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(z0) -> c8 C(c_c(c_c(z0))) -> c9(A(z0)) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: a_1, c_1, encArg_1, encode_a_1, encode_c_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_C_1, A_1, C_1 Compound Symbols: c1_2, c2_2, c3_2, c4_2, c5, c6, c7_4, c8, c9_1 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: A(z0) -> c5 C(z0) -> c8 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) ENCODE_A(z0) -> c3(A(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0)) A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: a_1, c_1, encArg_1, encode_a_1, encode_c_1 Defined Pair Symbols: ENCARG_1, ENCODE_A_1, ENCODE_C_1, A_1, C_1 Compound Symbols: c1_2, c2_2, c3_2, c4_2, c6, c7_4, c9_1 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_A(z0) -> c5(ENCARG(z0)) ENCODE_C(z0) -> c5(C(encArg(z0))) ENCODE_C(z0) -> c5(ENCARG(z0)) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: a_1, c_1, encArg_1, encode_a_1, encode_c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c2_2, c6, c7_4, c9_1, c5_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_A(z0) -> c5(ENCARG(z0)) ENCODE_C(z0) -> c5(ENCARG(z0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) encode_a(z0) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(z0) -> c5(C(encArg(z0))) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: a_1, c_1, encArg_1, encode_a_1, encode_c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c2_2, c6, c7_4, c9_1, c5_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_a(z0) -> a(encArg(z0)) encode_c(z0) -> c(encArg(z0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(z0) -> c5(C(encArg(z0))) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c2_2, c6, c7_4, c9_1, c5_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(z0)) -> c2(C(encArg(z0)), ENCARG(z0)) by ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(z0) -> c5(C(encArg(z0))) ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) S tuples: A(z0) -> c6 A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) C(c_c(c_c(z0))) -> c9(A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c7_4, c9_1, c5_1, c2_2 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(z0))) -> c7(A(c(a(a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(z0) -> c5(C(encArg(z0))) ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c7_4 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(z0) -> c5(C(encArg(z0))) by ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c7_4 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(cons_a(z0))) -> c2(C(a(encArg(z0))), ENCARG(cons_a(z0))) by ENCARG(cons_c(cons_a(x0))) -> c2(C(c_a(encArg(x0))), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(c_a(encArg(x0))), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c7_4, c2_1 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c7_4, c2_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCARG(cons_c(cons_c(z0))) -> c2(C(c(encArg(z0))), ENCARG(cons_c(z0))) by ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c7_4, c2_2, c2_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(x0))) -> c7(A(c_c(a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c7_4, c2_2, c2_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(x0))) -> c7(A(c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c7_4, c2_2, c2_1, c7_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(x0))) -> c7(A(c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c7_4, c2_2, c2_1, c7_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(z0))) -> c7(A(c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_a(x0))) -> c7(A(c_c(a(c_a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(c_a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c(c_a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c7_4, c2_2, c2_1, c7_2, c7_3 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) by A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c_c(a(a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(c_a(a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c(a(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(cons_a(z0)) -> c5(C(a(encArg(z0)))) by ENCODE_C(cons_a(x0)) -> c5(C(c_a(encArg(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(c_a(encArg(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: ENCODE_C(cons_a(x0)) -> c5(C(c_a(encArg(x0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace ENCODE_C(cons_c(z0)) -> c5(C(c(encArg(z0)))) by ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) by A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace A(c_a(c_a(x0))) -> c7(A(c(c_a(c_a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(z0))) -> c7(A(c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace A(c_a(c_a(x0))) -> c7(A(c(c_a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) by A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(cons_a(z0)) -> a(encArg(z0)) encArg(cons_c(z0)) -> c(encArg(z0)) a(z0) -> c_a(z0) a(z0) -> z0 a(c_a(c_a(z0))) -> a(c(a(a(z0)))) c(z0) -> c_c(z0) c(c_c(c_c(z0))) -> a(z0) Tuples: ENCARG(cons_a(z0)) -> c1(A(encArg(z0)), ENCARG(z0)) A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) ENCODE_A(z0) -> c5(A(encArg(z0))) ENCARG(cons_c(cons_a(x0))) -> c2(C(encArg(x0)), ENCARG(cons_a(x0))) ENCARG(cons_c(cons_a(cons_a(z0)))) -> c2(C(a(a(encArg(z0)))), ENCARG(cons_a(cons_a(z0)))) ENCARG(cons_c(cons_a(cons_c(z0)))) -> c2(C(a(c(encArg(z0)))), ENCARG(cons_a(cons_c(z0)))) ENCARG(cons_c(cons_a(x0))) -> c2(ENCARG(cons_a(x0))) ENCARG(cons_c(cons_c(x0))) -> c2(C(c_c(encArg(x0))), ENCARG(cons_c(x0))) ENCARG(cons_c(cons_c(cons_a(z0)))) -> c2(C(c(a(encArg(z0)))), ENCARG(cons_c(cons_a(z0)))) ENCARG(cons_c(cons_c(cons_c(z0)))) -> c2(C(c(c(encArg(z0)))), ENCARG(cons_c(cons_c(z0)))) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) ENCODE_C(cons_a(x0)) -> c5(C(encArg(x0))) ENCODE_C(cons_a(cons_a(z0))) -> c5(C(a(a(encArg(z0))))) ENCODE_C(cons_a(cons_c(z0))) -> c5(C(a(c(encArg(z0))))) ENCODE_C(cons_c(x0)) -> c5(C(c_c(encArg(x0)))) ENCODE_C(cons_c(cons_a(z0))) -> c5(C(c(a(encArg(z0))))) ENCODE_C(cons_c(cons_c(z0))) -> c5(C(c(c(encArg(z0))))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) S tuples: A(z0) -> c6 C(c_c(c_c(z0))) -> c9(A(z0)) A(c_a(c_a(x0))) -> c7(A(c_c(c_a(a(x0)))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(x0))) -> c7(A(c_c(a(x0))), C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(z0))) -> c7(A(c_c(a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c_c(a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(c_a(a(c(a(a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0))) A(c_a(c_a(z0))) -> c7(A(c(z0)), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(z0)))) -> c7(A(c(a(c(a(a(z0)))))), C(a(a(c_a(z0)))), A(a(c_a(z0))), A(c_a(z0))) A(c_a(c_a(x0))) -> c7(C(a(a(x0))), A(a(x0)), A(x0)) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(c_a(c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c_c(a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(c_a(a(x0))))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(A(c(a(a(c(a(x0)))))), C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(c_a(c_a(z0))))) -> c7(A(c(a(a(c(a(c_a(z0))))))), C(a(a(c_a(c_a(z0))))), A(a(c_a(c_a(z0)))), A(c_a(c_a(z0)))) A(c_a(c_a(c_a(c_a(c_a(c_a(z0))))))) -> c7(A(c(a(a(c(a(a(c(a(a(z0)))))))))), C(a(a(c_a(c_a(c_a(c_a(z0))))))), A(a(c_a(c_a(c_a(c_a(z0)))))), A(c_a(c_a(c_a(c_a(z0)))))) A(c_a(c_a(c_a(c_a(x0))))) -> c7(C(a(a(c_a(c_a(x0))))), A(a(c_a(c_a(x0)))), A(c_a(c_a(x0)))) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(c_a(z0)))), C(a(a(z0))), A(a(z0)), A(z0)) A(c_a(c_a(z0))) -> c7(A(c_c(c_a(z0))), C(a(a(z0))), A(a(z0)), A(z0)) K tuples:none Defined Rule Symbols: encArg_1, a_1, c_1 Defined Pair Symbols: ENCARG_1, A_1, C_1, ENCODE_A_1, ENCODE_C_1 Compound Symbols: c1_2, c6, c9_1, c5_1, c2_2, c2_1, c7_4, c7_2, c7_3