/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- KILLED proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 150 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 237 ms] (12) BOUNDS(1, INF) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxTRS (17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (18) CpxRelTRS (19) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRelTRS (21) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxWeightedTrs (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxTypedWeightedTrs (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxRNTS (33) CompletionProof [UPPER BOUND(ID), 0 ms] (34) CpxTypedWeightedCompleteTrs (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (40) CdtProblem (41) 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), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtGraphSplitRhsProof [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) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (76) CdtProblem (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) 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: c(c(c(y))) -> c(c(a(y, 0))) c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 ---------------------------------------- (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: c(c(c(y))) -> c(c(a(y, 0))) c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 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: c(c(c(y))) -> c(c(a(y, 0))) c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(c(c(y))) -> c(c(a(y, 0'))) c(a(a(0', x), y)) -> a(c(c(c(0'))), y) c(y) -> y The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0') -> 0' encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0' Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: c(c(c(y))) -> c(c(a(y, 0'))) c(a(a(0', x), y)) -> a(c(c(c(0'))), y) c(y) -> y encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0') -> 0' encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0' Types: c :: 0':a:cons_c -> 0':a:cons_c a :: 0':a:cons_c -> 0':a:cons_c -> 0':a:cons_c 0' :: 0':a:cons_c encArg :: 0':a:cons_c -> 0':a:cons_c cons_c :: 0':a:cons_c -> 0':a:cons_c encode_c :: 0':a:cons_c -> 0':a:cons_c encode_a :: 0':a:cons_c -> 0':a:cons_c -> 0':a:cons_c encode_0 :: 0':a:cons_c hole_0':a:cons_c1_3 :: 0':a:cons_c gen_0':a:cons_c2_3 :: Nat -> 0':a:cons_c ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: c, encArg They will be analysed ascendingly in the following order: c < encArg ---------------------------------------- (10) Obligation: TRS: Rules: c(c(c(y))) -> c(c(a(y, 0'))) c(a(a(0', x), y)) -> a(c(c(c(0'))), y) c(y) -> y encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0') -> 0' encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0' Types: c :: 0':a:cons_c -> 0':a:cons_c a :: 0':a:cons_c -> 0':a:cons_c -> 0':a:cons_c 0' :: 0':a:cons_c encArg :: 0':a:cons_c -> 0':a:cons_c cons_c :: 0':a:cons_c -> 0':a:cons_c encode_c :: 0':a:cons_c -> 0':a:cons_c encode_a :: 0':a:cons_c -> 0':a:cons_c -> 0':a:cons_c encode_0 :: 0':a:cons_c hole_0':a:cons_c1_3 :: 0':a:cons_c gen_0':a:cons_c2_3 :: Nat -> 0':a:cons_c Generator Equations: gen_0':a:cons_c2_3(0) <=> 0' gen_0':a:cons_c2_3(+(x, 1)) <=> a(gen_0':a:cons_c2_3(x), 0') The following defined symbols remain to be analysed: c, encArg They will be analysed ascendingly in the following order: c < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':a:cons_c2_3(n69_3)) -> gen_0':a:cons_c2_3(n69_3), rt in Omega(0) Induction Base: encArg(gen_0':a:cons_c2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':a:cons_c2_3(+(n69_3, 1))) ->_R^Omega(0) a(encArg(gen_0':a:cons_c2_3(n69_3)), encArg(0')) ->_IH a(gen_0':a:cons_c2_3(c70_3), encArg(0')) ->_R^Omega(0) a(gen_0':a:cons_c2_3(n69_3), 0') We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (12) BOUNDS(1, INF) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(c(c(y))) -> c(c(a(y, 0))) c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 Rewrite Strategy: FULL ---------------------------------------- (15) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: c(c(c(y))) -> c(c(a(y, 0))) c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 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: c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y c(c_c(c_c(y))) -> c(c(a(y, 0))) The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 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: c(a(a(0, x), y)) -> a(c(c(c(0))), y) c(y) -> y c(c_c(c_c(y))) -> c(c(a(y, 0))) The (relative) TRS S consists of the following rules: encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 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: c(a(a(0, x), y)) -> a(c(c(c(0))), y) [1] c(y) -> y [1] c(c_c(c_c(y))) -> c(c(a(y, 0))) [1] encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [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: c(a(a(0, x), y)) -> a(c(c(c(0))), y) [1] c(y) -> y [1] c(c_c(c_c(y))) -> c(c(a(y, 0))) [1] encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] c(x0) -> c_c(x0) [0] The TRS has the following type information: c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c 0 :: 0:a:c_c:cons_c c_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encArg :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c cons_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_0 :: 0:a:c_c:cons_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_c_1 encode_a_2 encode_0 c_1 Due to the following rules being added: encArg(v0) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] c(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (26) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: c(a(a(0, x), y)) -> a(c(c(c(0))), y) [1] c(y) -> y [1] c(c_c(c_c(y))) -> c(c(a(y, 0))) [1] encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] c(x0) -> c_c(x0) [0] encArg(v0) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] c(v0) -> 0 [0] The TRS has the following type information: c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c 0 :: 0:a:c_c:cons_c c_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encArg :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c cons_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_0 :: 0:a:c_c: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: c(a(a(0, x), y)) -> a(c(c(0)), y) [2] c(a(a(0, x), y)) -> a(c(c(c_c(0))), y) [1] c(a(a(0, x), y)) -> a(c(c(0)), y) [1] c(y) -> y [1] c(c_c(c_c(a(0, x')))) -> c(a(c(c(c(0))), 0)) [2] c(c_c(c_c(y))) -> c(a(y, 0)) [2] c(c_c(c_c(y))) -> c(c_c(a(y, 0))) [1] c(c_c(c_c(y))) -> c(0) [1] encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(cons_c(a(x_1', x_2'))) -> c(a(encArg(x_1'), encArg(x_2'))) [0] encArg(cons_c(0)) -> c(0) [0] encArg(cons_c(cons_c(x_1''))) -> c(c(encArg(x_1''))) [0] encArg(cons_c(x_1)) -> c(0) [0] encode_c(a(x_11, x_2'')) -> c(a(encArg(x_11), encArg(x_2''))) [0] encode_c(0) -> c(0) [0] encode_c(cons_c(x_12)) -> c(c(encArg(x_12))) [0] encode_c(x_1) -> c(0) [0] encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] c(x0) -> c_c(x0) [0] encArg(v0) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] c(v0) -> 0 [0] The TRS has the following type information: c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c 0 :: 0:a:c_c:cons_c c_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encArg :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c cons_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_c :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_a :: 0:a:c_c:cons_c -> 0:a:c_c:cons_c -> 0:a:c_c:cons_c encode_0 :: 0:a:c_c: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: 0 => 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: c(z) -{ 1 }-> y :|: y >= 0, z = y c(z) -{ 1 }-> c(0) :|: z = 1 + (1 + y), y >= 0 c(z) -{ 1 }-> c(1 + (1 + y + 0)) :|: z = 1 + (1 + y), y >= 0 c(z) -{ 2 }-> c(1 + y + 0) :|: z = 1 + (1 + y), y >= 0 c(z) -{ 2 }-> c(1 + c(c(c(0))) + 0) :|: x' >= 0, z = 1 + (1 + (1 + 0 + x')) c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 2 }-> 1 + c(c(0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 c(z) -{ 1 }-> 1 + c(c(0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 c(z) -{ 1 }-> 1 + c(c(1 + 0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 encArg(z) -{ 0 }-> c(c(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> c(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> c(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> c(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_a(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_c(z) -{ 0 }-> c(c(encArg(x_12))) :|: z = 1 + x_12, x_12 >= 0 encode_c(z) -{ 0 }-> c(0) :|: z = 0 encode_c(z) -{ 0 }-> c(0) :|: x_1 >= 0, z = x_1 encode_c(z) -{ 0 }-> c(1 + encArg(x_11) + encArg(x_2'')) :|: z = 1 + x_11 + x_2'', x_11 >= 0, x_2'' >= 0 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: c(z) -{ 1 }-> z :|: z >= 0 c(z) -{ 1 }-> c(0) :|: z - 2 >= 0 c(z) -{ 1 }-> c(1 + (1 + (z - 2) + 0)) :|: z - 2 >= 0 c(z) -{ 2 }-> c(1 + c(c(c(0))) + 0) :|: z - 3 >= 0 c(z) -{ 2 }-> c(1 + (z - 2) + 0) :|: z - 2 >= 0 c(z) -{ 0 }-> 0 :|: z >= 0 c(z) -{ 0 }-> 1 + z :|: z >= 0 c(z) -{ 2 }-> 1 + c(c(0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 c(z) -{ 1 }-> 1 + c(c(0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 c(z) -{ 1 }-> 1 + c(c(1 + 0)) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 encArg(z) -{ 0 }-> c(c(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> c(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> c(0) :|: z - 1 >= 0 encArg(z) -{ 0 }-> c(1 + encArg(x_1') + encArg(x_2')) :|: z = 1 + (1 + x_1' + x_2'), x_2' >= 0, x_1' >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_a(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0 encode_c(z) -{ 0 }-> c(c(encArg(z - 1))) :|: z - 1 >= 0 encode_c(z) -{ 0 }-> c(0) :|: z = 0 encode_c(z) -{ 0 }-> c(0) :|: z >= 0 encode_c(z) -{ 0 }-> c(1 + encArg(x_11) + encArg(x_2'')) :|: z = 1 + x_11 + x_2'', x_11 >= 0, x_2'' >= 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_c(v0) -> null_encode_c [0] encode_a(v0, v1) -> null_encode_a [0] encode_0 -> null_encode_0 [0] c(v0) -> null_c [0] And the following fresh constants: null_encArg, null_encode_c, null_encode_a, null_encode_0, null_c ---------------------------------------- (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: c(a(a(0, x), y)) -> a(c(c(c(0))), y) [1] c(y) -> y [1] c(c_c(c_c(y))) -> c(c(a(y, 0))) [1] encArg(a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(cons_c(x_1)) -> c(encArg(x_1)) [0] encode_c(x_1) -> c(encArg(x_1)) [0] encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] c(x0) -> c_c(x0) [0] encArg(v0) -> null_encArg [0] encode_c(v0) -> null_encode_c [0] encode_a(v0, v1) -> null_encode_a [0] encode_0 -> null_encode_0 [0] c(v0) -> null_c [0] The TRS has the following type information: c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c a :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c 0 :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c c_c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c encArg :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c cons_c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c encode_c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c encode_a :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c -> 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c encode_0 :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c null_encArg :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c null_encode_c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c null_encode_a :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c null_encode_0 :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_c null_c :: 0:a:c_c:cons_c:null_encArg:null_encode_c:null_encode_a:null_encode_0:null_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: 0 => 0 null_encArg => 0 null_encode_c => 0 null_encode_a => 0 null_encode_0 => 0 null_c => 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: c(z) -{ 1 }-> y :|: y >= 0, z = y c(z) -{ 1 }-> c(c(1 + y + 0)) :|: z = 1 + (1 + y), y >= 0 c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 c(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 c(z) -{ 1 }-> 1 + c(c(c(0))) + y :|: z = 1 + (1 + 0 + x) + y, x >= 0, y >= 0 encArg(z) -{ 0 }-> c(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_a(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 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(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(0) -> c2 ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0)) ENCODE_A(z0, z1) -> c5(ENCARG(z0), ENCARG(z1)) ENCODE_0 -> c6 C(z0) -> c7 C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) S tuples: C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) K tuples:none Defined Rule Symbols: c_1, encArg_1, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_C_1, ENCODE_A_2, ENCODE_0, C_1 Compound Symbols: c1_2, c2, c3_2, c4_2, c5_2, c6, c7, c8_3, c9, c10_2 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_A(z0, z1) -> c5(ENCARG(z0), ENCARG(z1)) Removed 3 trailing nodes: ENCODE_0 -> c6 ENCARG(0) -> c2 C(z0) -> c7 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) ENCODE_C(z0) -> c4(C(encArg(z0)), ENCARG(z0)) C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) S tuples: C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) K tuples:none Defined Rule Symbols: c_1, encArg_1, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_C_1, C_1 Compound Symbols: c1_2, c3_2, c4_2, c8_3, c9, c10_2 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) ENCODE_C(z0) -> c2(C(encArg(z0))) ENCODE_C(z0) -> c2(ENCARG(z0)) S tuples: C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) K tuples:none Defined Rule Symbols: c_1, encArg_1, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c8_3, c9, c10_2, c2_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_C(z0) -> c2(ENCARG(z0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) ENCODE_C(z0) -> c2(C(encArg(z0))) S tuples: C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) K tuples:none Defined Rule Symbols: c_1, encArg_1, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c8_3, c9, c10_2, c2_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) ENCODE_C(z0) -> c2(C(encArg(z0))) S tuples: C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c8_3, c9, c10_2, c2_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, z0), z1)) -> c8(C(c(c(0))), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) ENCODE_C(z0) -> c2(C(encArg(z0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c10_2, c2_1, c8_3 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c_c(c_c(z0))) -> c10(C(c(a(z0, 0))), C(a(z0, 0))) by C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) S tuples: C(z0) -> c9 C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c8_3, c10_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c_c(c(0))), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) S tuples: C(z0) -> c9 C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c8_3, c10_2, c8_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c(0)), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) S tuples: C(z0) -> c9 C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c8_3, c10_2, c8_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c(c_c(0))), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(c_c(c_c(a(0, z0)))) -> c10(C(a(c(c(c(0))), 0)), C(a(a(0, z0), 0))) by C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1 ---------------------------------------- (61) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c(0))) by C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c8(C(0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1 ---------------------------------------- (67) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1, c5_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(0), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(0), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0), C(0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0), C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1, c5_1 ---------------------------------------- (71) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0))) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_1, c8_2, c10_1, c4_1, c5_1, c6_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c(0))) by C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c8(C(0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_2, c10_1, c8_1, c4_1, c5_1, c6_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_2, c10_1, c8_1, c4_1, c5_1, c6_1 ---------------------------------------- (77) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c8_2, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace C(a(a(0, x0), x1)) -> c8(C(c(0)), C(0)) by C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0)), C(0)) C(a(a(0, x0), x1)) -> c8(C(0), C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1, c8_2 ---------------------------------------- (81) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c1_2, c3_2, c9, c2_1, c10_2, c8_3, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1, c11_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(z0, z1)) -> c1(ENCARG(z0), ENCARG(z1)) by ENCARG(a(a(y0, y1), z1)) -> c1(ENCARG(a(y0, y1)), ENCARG(z1)) ENCARG(a(z0, a(y0, y1))) -> c1(ENCARG(z0), ENCARG(a(y0, y1))) ENCARG(a(cons_c(y0), z1)) -> c1(ENCARG(cons_c(y0)), ENCARG(z1)) ENCARG(a(z0, cons_c(y0))) -> c1(ENCARG(z0), ENCARG(cons_c(y0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) ENCARG(a(a(y0, y1), z1)) -> c1(ENCARG(a(y0, y1)), ENCARG(z1)) ENCARG(a(z0, a(y0, y1))) -> c1(ENCARG(z0), ENCARG(a(y0, y1))) ENCARG(a(cons_c(y0), z1)) -> c1(ENCARG(cons_c(y0)), ENCARG(z1)) ENCARG(a(z0, cons_c(y0))) -> c1(ENCARG(z0), ENCARG(cons_c(y0))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c3_2, c9, c2_1, c10_2, c8_3, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1, c11_1, c1_2 ---------------------------------------- (85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(a(y0, y1), z1)) -> c1(ENCARG(a(y0, y1)), ENCARG(z1)) by ENCARG(a(a(z0, z1), cons_c(y0))) -> c1(ENCARG(a(z0, z1)), ENCARG(cons_c(y0))) ENCARG(a(a(a(y0, y1), z1), z2)) -> c1(ENCARG(a(a(y0, y1), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(a(y0, y1), y2))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(a(y0, y1), y2))) ENCARG(a(a(z0, a(y1, y2)), z2)) -> c1(ENCARG(a(z0, a(y1, y2))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, a(y1, y2)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, a(y1, y2)))) ENCARG(a(a(cons_c(y0), z1), z2)) -> c1(ENCARG(a(cons_c(y0), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(cons_c(y0), y1))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(cons_c(y0), y1))) ENCARG(a(a(z0, cons_c(y1)), z2)) -> c1(ENCARG(a(z0, cons_c(y1))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, cons_c(y1)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, cons_c(y1)))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) ENCARG(a(z0, a(y0, y1))) -> c1(ENCARG(z0), ENCARG(a(y0, y1))) ENCARG(a(cons_c(y0), z1)) -> c1(ENCARG(cons_c(y0)), ENCARG(z1)) ENCARG(a(z0, cons_c(y0))) -> c1(ENCARG(z0), ENCARG(cons_c(y0))) ENCARG(a(a(z0, z1), cons_c(y0))) -> c1(ENCARG(a(z0, z1)), ENCARG(cons_c(y0))) ENCARG(a(a(a(y0, y1), z1), z2)) -> c1(ENCARG(a(a(y0, y1), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(a(y0, y1), y2))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(a(y0, y1), y2))) ENCARG(a(a(z0, a(y1, y2)), z2)) -> c1(ENCARG(a(z0, a(y1, y2))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, a(y1, y2)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, a(y1, y2)))) ENCARG(a(a(cons_c(y0), z1), z2)) -> c1(ENCARG(a(cons_c(y0), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(cons_c(y0), y1))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(cons_c(y0), y1))) ENCARG(a(a(z0, cons_c(y1)), z2)) -> c1(ENCARG(a(z0, cons_c(y1))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, cons_c(y1)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, cons_c(y1)))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c3_2, c9, c2_1, c10_2, c8_3, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1, c11_1, c1_2 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(a(z0, a(y0, y1))) -> c1(ENCARG(z0), ENCARG(a(y0, y1))) by ENCARG(a(cons_c(y0), a(z1, z2))) -> c1(ENCARG(cons_c(y0)), ENCARG(a(z1, z2))) ENCARG(a(a(y0, a(y1, y2)), a(z1, z2))) -> c1(ENCARG(a(y0, a(y1, y2))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(z1, a(y1, y2)))) -> c1(ENCARG(z0), ENCARG(a(z1, a(y1, y2)))) ENCARG(a(a(cons_c(y0), y1), a(z1, z2))) -> c1(ENCARG(a(cons_c(y0), y1)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(cons_c(y0), z2))) -> c1(ENCARG(z0), ENCARG(a(cons_c(y0), z2))) ENCARG(a(a(y0, cons_c(y1)), a(z1, z2))) -> c1(ENCARG(a(y0, cons_c(y1))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(z1, cons_c(y1)))) -> c1(ENCARG(z0), ENCARG(a(z1, cons_c(y1)))) ENCARG(a(a(a(y0, y1), cons_c(y2)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), cons_c(y2))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), cons_c(y2)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), cons_c(y2)))) ENCARG(a(a(a(a(y0, y1), y2), y3), a(z1, z2))) -> c1(ENCARG(a(a(a(y0, y1), y2), y3)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(a(y0, y1), y2), z2))) -> c1(ENCARG(z0), ENCARG(a(a(a(y0, y1), y2), z2))) ENCARG(a(a(a(y0, y1), a(a(y2, y3), y4)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(a(y2, y3), y4))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(a(y2, y3), y4)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(a(y2, y3), y4)))) ENCARG(a(a(a(y0, a(y1, y2)), y3), a(z1, z2))) -> c1(ENCARG(a(a(y0, a(y1, y2)), y3)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, a(y1, y2)), z2))) -> c1(ENCARG(z0), ENCARG(a(a(y0, a(y1, y2)), z2))) ENCARG(a(a(a(y0, y1), a(y2, a(y3, y4))), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(y2, a(y3, y4)))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(y2, a(y3, y4))))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(y2, a(y3, y4))))) ENCARG(a(a(a(cons_c(y0), y1), y2), a(z1, z2))) -> c1(ENCARG(a(a(cons_c(y0), y1), y2)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(cons_c(y0), y1), z2))) -> c1(ENCARG(z0), ENCARG(a(a(cons_c(y0), y1), z2))) ENCARG(a(a(a(y0, y1), a(cons_c(y2), y3)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(cons_c(y2), y3))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(cons_c(y2), y3)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(cons_c(y2), y3)))) ENCARG(a(a(a(y0, cons_c(y1)), y2), a(z1, z2))) -> c1(ENCARG(a(a(y0, cons_c(y1)), y2)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, cons_c(y1)), z2))) -> c1(ENCARG(z0), ENCARG(a(a(y0, cons_c(y1)), z2))) ENCARG(a(a(a(y0, y1), a(y2, cons_c(y3))), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(y2, cons_c(y3)))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(y2, cons_c(y3))))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(y2, cons_c(y3))))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: encArg(a(z0, z1)) -> a(encArg(z0), encArg(z1)) encArg(0) -> 0 encArg(cons_c(z0)) -> c(encArg(z0)) c(z0) -> c_c(z0) c(a(a(0, z0), z1)) -> a(c(c(c(0))), z1) c(z0) -> z0 c(c_c(c_c(z0))) -> c(c(a(z0, 0))) Tuples: ENCARG(cons_c(z0)) -> c3(C(encArg(z0)), ENCARG(z0)) C(z0) -> c9 ENCODE_C(z0) -> c2(C(encArg(z0))) C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c5(C(c_c(0))) C(a(a(0, x0), x1)) -> c5(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(c_c(0))) C(a(a(0, x0), x1)) -> c7(C(0)) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) ENCARG(a(cons_c(y0), z1)) -> c1(ENCARG(cons_c(y0)), ENCARG(z1)) ENCARG(a(z0, cons_c(y0))) -> c1(ENCARG(z0), ENCARG(cons_c(y0))) ENCARG(a(a(z0, z1), cons_c(y0))) -> c1(ENCARG(a(z0, z1)), ENCARG(cons_c(y0))) ENCARG(a(a(a(y0, y1), z1), z2)) -> c1(ENCARG(a(a(y0, y1), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(a(y0, y1), y2))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(a(y0, y1), y2))) ENCARG(a(a(z0, a(y1, y2)), z2)) -> c1(ENCARG(a(z0, a(y1, y2))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, a(y1, y2)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, a(y1, y2)))) ENCARG(a(a(cons_c(y0), z1), z2)) -> c1(ENCARG(a(cons_c(y0), z1)), ENCARG(z2)) ENCARG(a(a(z0, z1), a(cons_c(y0), y1))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(cons_c(y0), y1))) ENCARG(a(a(z0, cons_c(y1)), z2)) -> c1(ENCARG(a(z0, cons_c(y1))), ENCARG(z2)) ENCARG(a(a(z0, z1), a(y0, cons_c(y1)))) -> c1(ENCARG(a(z0, z1)), ENCARG(a(y0, cons_c(y1)))) ENCARG(a(cons_c(y0), a(z1, z2))) -> c1(ENCARG(cons_c(y0)), ENCARG(a(z1, z2))) ENCARG(a(a(y0, a(y1, y2)), a(z1, z2))) -> c1(ENCARG(a(y0, a(y1, y2))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(z1, a(y1, y2)))) -> c1(ENCARG(z0), ENCARG(a(z1, a(y1, y2)))) ENCARG(a(a(cons_c(y0), y1), a(z1, z2))) -> c1(ENCARG(a(cons_c(y0), y1)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(cons_c(y0), z2))) -> c1(ENCARG(z0), ENCARG(a(cons_c(y0), z2))) ENCARG(a(a(y0, cons_c(y1)), a(z1, z2))) -> c1(ENCARG(a(y0, cons_c(y1))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(z1, cons_c(y1)))) -> c1(ENCARG(z0), ENCARG(a(z1, cons_c(y1)))) ENCARG(a(a(a(y0, y1), cons_c(y2)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), cons_c(y2))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), cons_c(y2)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), cons_c(y2)))) ENCARG(a(a(a(a(y0, y1), y2), y3), a(z1, z2))) -> c1(ENCARG(a(a(a(y0, y1), y2), y3)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(a(y0, y1), y2), z2))) -> c1(ENCARG(z0), ENCARG(a(a(a(y0, y1), y2), z2))) ENCARG(a(a(a(y0, y1), a(a(y2, y3), y4)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(a(y2, y3), y4))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(a(y2, y3), y4)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(a(y2, y3), y4)))) ENCARG(a(a(a(y0, a(y1, y2)), y3), a(z1, z2))) -> c1(ENCARG(a(a(y0, a(y1, y2)), y3)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, a(y1, y2)), z2))) -> c1(ENCARG(z0), ENCARG(a(a(y0, a(y1, y2)), z2))) ENCARG(a(a(a(y0, y1), a(y2, a(y3, y4))), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(y2, a(y3, y4)))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(y2, a(y3, y4))))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(y2, a(y3, y4))))) ENCARG(a(a(a(cons_c(y0), y1), y2), a(z1, z2))) -> c1(ENCARG(a(a(cons_c(y0), y1), y2)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(cons_c(y0), y1), z2))) -> c1(ENCARG(z0), ENCARG(a(a(cons_c(y0), y1), z2))) ENCARG(a(a(a(y0, y1), a(cons_c(y2), y3)), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(cons_c(y2), y3))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(cons_c(y2), y3)))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(cons_c(y2), y3)))) ENCARG(a(a(a(y0, cons_c(y1)), y2), a(z1, z2))) -> c1(ENCARG(a(a(y0, cons_c(y1)), y2)), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, cons_c(y1)), z2))) -> c1(ENCARG(z0), ENCARG(a(a(y0, cons_c(y1)), z2))) ENCARG(a(a(a(y0, y1), a(y2, cons_c(y3))), a(z1, z2))) -> c1(ENCARG(a(a(y0, y1), a(y2, cons_c(y3)))), ENCARG(a(z1, z2))) ENCARG(a(z0, a(a(y0, y1), a(y2, cons_c(y3))))) -> c1(ENCARG(z0), ENCARG(a(a(y0, y1), a(y2, cons_c(y3))))) S tuples: C(z0) -> c9 C(c_c(c_c(x0))) -> c10(C(c_c(a(x0, 0))), C(a(x0, 0))) C(c_c(c_c(x0))) -> c10(C(a(x0, 0)), C(a(x0, 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(c_c(0))), C(c(0)), C(0)) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c_c(c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(0)), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c_c(c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(c(c(c_c(0))), 0)), C(a(a(0, x0), 0))) C(c_c(c_c(a(0, x0)))) -> c10(C(a(a(0, x0), 0))) C(a(a(0, x0), x1)) -> c8(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(c_c(0))) C(a(a(0, x0), x1)) -> c4(C(0)) C(a(a(0, x0), x1)) -> c8(C(0)) C(a(a(0, x0), x1)) -> c6(C(0)) C(a(a(0, x0), x1)) -> c6(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(c_c(0))) C(a(a(0, x0), x1)) -> c11(C(0)) K tuples:none Defined Rule Symbols: encArg_1, c_1 Defined Pair Symbols: ENCARG_1, C_1, ENCODE_C_1 Compound Symbols: c3_2, c9, c2_1, c10_2, c8_3, c10_1, c8_1, c4_1, c5_1, c6_1, c7_1, c11_1, c1_2