/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, INF).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), 153 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), 282 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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CompletionProof [UPPER BOUND(ID), 0 ms] (30) CpxTypedWeightedCompleteTrs (31) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) SimplificationProof [BOTH BOUNDS(ID, 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 21 ms] (66) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) 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(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0', x))) a(y, x) -> y a(y, c(b(a(0', x), 0'))) -> b(a(c(b(0', y)), x), 0') The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0') -> 0' encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0', x))) a(y, x) -> y a(y, c(b(a(0', x), 0'))) -> b(a(c(b(0', y)), x), 0') encArg(c(x_1)) -> c(encArg(x_1)) encArg(0') -> 0' encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a c :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a 0' :: c:0':cons_b:cons_a encArg :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a cons_b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a cons_a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_c :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_0 :: c:0':cons_b:cons_a hole_c:0':cons_b:cons_a1_3 :: c:0':cons_b:cons_a gen_c:0':cons_b:cons_a2_3 :: Nat -> c:0':cons_b:cons_a ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: b, a, encArg They will be analysed ascendingly in the following order: b = a b < encArg a < encArg ---------------------------------------- (10) Obligation: TRS: Rules: b(x, y) -> c(a(c(y), a(0', x))) a(y, x) -> y a(y, c(b(a(0', x), 0'))) -> b(a(c(b(0', y)), x), 0') encArg(c(x_1)) -> c(encArg(x_1)) encArg(0') -> 0' encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a c :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a 0' :: c:0':cons_b:cons_a encArg :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a cons_b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a cons_a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_b :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_c :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_a :: c:0':cons_b:cons_a -> c:0':cons_b:cons_a -> c:0':cons_b:cons_a encode_0 :: c:0':cons_b:cons_a hole_c:0':cons_b:cons_a1_3 :: c:0':cons_b:cons_a gen_c:0':cons_b:cons_a2_3 :: Nat -> c:0':cons_b:cons_a Generator Equations: gen_c:0':cons_b:cons_a2_3(0) <=> 0' gen_c:0':cons_b:cons_a2_3(+(x, 1)) <=> c(gen_c:0':cons_b:cons_a2_3(x)) The following defined symbols remain to be analysed: a, b, encArg They will be analysed ascendingly in the following order: b = a b < encArg a < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_c:0':cons_b:cons_a2_3(n119_3)) -> gen_c:0':cons_b:cons_a2_3(n119_3), rt in Omega(0) Induction Base: encArg(gen_c:0':cons_b:cons_a2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_c:0':cons_b:cons_a2_3(+(n119_3, 1))) ->_R^Omega(0) c(encArg(gen_c:0':cons_b:cons_a2_3(n119_3))) ->_IH c(gen_c:0':cons_b:cons_a2_3(c120_3)) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) 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: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 a(x0, x1) -> c_a(x0, x1) b(x0, x1) -> c_b(x0, x1) Rewrite Strategy: FULL ---------------------------------------- (19) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(x, y) -> c(a(c(y), a(0, x))) a(y, x) -> y a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(0) -> 0 encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2)) encode_0 -> 0 a(x0, x1) -> c_a(x0, x1) b(x0, x1) -> c_b(x0, x1) Rewrite Strategy: INNERMOST ---------------------------------------- (21) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (22) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: b(x, y) -> c(a(c(y), a(0, x))) [1] a(y, x) -> y [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) [1] encArg(c(x_1)) -> c(encArg(x_1)) [0] encArg(0) -> 0 [0] encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) [0] encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) [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] a(x0, x1) -> c_a(x0, x1) [0] b(x0, x1) -> c_b(x0, x1) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (23) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (24) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(x, y) -> c(a(c(y), a(0, x))) [1] a(y, x) -> y [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) [1] encArg(c(x_1)) -> c(encArg(x_1)) [0] encArg(0) -> 0 [0] encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) [0] encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) [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] a(x0, x1) -> c_a(x0, x1) [0] b(x0, x1) -> c_b(x0, x1) [0] The TRS has the following type information: b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a 0 :: c:0:c_a:c_b:cons_b:cons_a c_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encArg :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_0 :: c:0:c_a:c_b:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (25) 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_b(v0, v1) -> null_encode_b [0] encode_c(v0) -> null_encode_c [0] encode_a(v0, v1) -> null_encode_a [0] encode_0 -> null_encode_0 [0] a(v0, v1) -> null_a [0] b(v0, v1) -> null_b [0] And the following fresh constants: null_encArg, null_encode_b, null_encode_c, null_encode_a, null_encode_0, null_a, null_b ---------------------------------------- (26) 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: b(x, y) -> c(a(c(y), a(0, x))) [1] a(y, x) -> y [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) [1] encArg(c(x_1)) -> c(encArg(x_1)) [0] encArg(0) -> 0 [0] encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) [0] encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) [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] a(x0, x1) -> c_a(x0, x1) [0] b(x0, x1) -> c_b(x0, x1) [0] encArg(v0) -> null_encArg [0] encode_b(v0, v1) -> null_encode_b [0] encode_c(v0) -> null_encode_c [0] encode_a(v0, v1) -> null_encode_a [0] encode_0 -> null_encode_0 [0] a(v0, v1) -> null_a [0] b(v0, v1) -> null_b [0] The TRS has the following type information: b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b c :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b 0 :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b c_b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b c_a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b encArg :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b cons_b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b cons_a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b encode_b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b encode_c :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b encode_a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b -> c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b encode_0 :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_encArg :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_encode_b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_encode_c :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_encode_a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_encode_0 :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_a :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b null_b :: c:0:c_a:c_b:cons_b:cons_a:null_encArg:null_encode_b:null_encode_c:null_encode_a:null_encode_0:null_a:null_b Rewrite Strategy: INNERMOST ---------------------------------------- (27) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_encArg => 0 null_encode_b => 0 null_encode_c => 0 null_encode_a => 0 null_encode_0 => 0 null_a => 0 null_b => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> y :|: z' = x, y >= 0, x >= 0, z = y a(z, z') -{ 1 }-> b(a(1 + b(0, y), x), 0) :|: z' = 1 + (1 + (1 + 0 + x) + 0), y >= 0, x >= 0, z = y a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 b(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 b(z, z') -{ 1 }-> 1 + a(1 + y, a(0, x)) :|: x >= 0, y >= 0, z = x, z' = y b(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> b(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> a(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_b(z, z') -{ 0 }-> b(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_b(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) 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_b_2 encode_c_1 encode_a_2 encode_0 a_2 b_2 Due to the following rules being added: encArg(v0) -> 0 [0] encode_b(v0, v1) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] a(v0, v1) -> 0 [0] b(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (30) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: b(x, y) -> c(a(c(y), a(0, x))) [1] a(y, x) -> y [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(b(0, y)), x), 0) [1] encArg(c(x_1)) -> c(encArg(x_1)) [0] encArg(0) -> 0 [0] encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) [0] encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) [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] a(x0, x1) -> c_a(x0, x1) [0] b(x0, x1) -> c_b(x0, x1) [0] encArg(v0) -> 0 [0] encode_b(v0, v1) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] a(v0, v1) -> 0 [0] b(v0, v1) -> 0 [0] The TRS has the following type information: b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a 0 :: c:0:c_a:c_b:cons_b:cons_a c_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encArg :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_0 :: c:0:c_a:c_b:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (31) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (32) 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: b(x, y) -> c(a(c(y), 0)) [2] b(c(c_b(c_a(0, x'), 0)), y) -> c(a(c(y), b(a(c(b(0, 0)), x'), 0))) [2] b(x, y) -> c(a(c(y), c_a(0, x))) [1] b(x, y) -> c(a(c(y), 0)) [1] a(y, x) -> y [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(c(a(c(y), a(0, 0)))), x), 0) [2] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(c_b(0, y)), x), 0) [1] a(y, c(c_b(c_a(0, x), 0))) -> b(a(c(0), x), 0) [1] encArg(c(x_1)) -> c(encArg(x_1)) [0] encArg(0) -> 0 [0] encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) [0] encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2)) [0] encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) [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] a(x0, x1) -> c_a(x0, x1) [0] b(x0, x1) -> c_b(x0, x1) [0] encArg(v0) -> 0 [0] encode_b(v0, v1) -> 0 [0] encode_c(v0) -> 0 [0] encode_a(v0, v1) -> 0 [0] encode_0 -> 0 [0] a(v0, v1) -> 0 [0] b(v0, v1) -> 0 [0] The TRS has the following type information: b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a 0 :: c:0:c_a:c_b:cons_b:cons_a c_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a c_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encArg :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a cons_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_b :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_c :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_a :: c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a -> c:0:c_a:c_b:cons_b:cons_a encode_0 :: c:0:c_a:c_b:cons_b:cons_a Rewrite Strategy: INNERMOST ---------------------------------------- (33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> y :|: z' = x, y >= 0, x >= 0, z = y a(z, z') -{ 1 }-> b(a(1 + 0, x), 0) :|: z' = 1 + (1 + (1 + 0 + x) + 0), y >= 0, x >= 0, z = y a(z, z') -{ 2 }-> b(a(1 + (1 + a(1 + y, a(0, 0))), x), 0) :|: z' = 1 + (1 + (1 + 0 + x) + 0), y >= 0, x >= 0, z = y a(z, z') -{ 1 }-> b(a(1 + (1 + 0 + y), x), 0) :|: z' = 1 + (1 + (1 + 0 + x) + 0), y >= 0, x >= 0, z = y a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 b(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 b(z, z') -{ 2 }-> 1 + a(1 + y, b(a(1 + b(0, 0), x'), 0)) :|: z = 1 + (1 + (1 + 0 + x') + 0), x' >= 0, y >= 0, z' = y b(z, z') -{ 2 }-> 1 + a(1 + y, 0) :|: x >= 0, y >= 0, z = x, z' = y b(z, z') -{ 1 }-> 1 + a(1 + y, 0) :|: x >= 0, y >= 0, z = x, z' = y b(z, z') -{ 1 }-> 1 + a(1 + y, 1 + 0 + x) :|: x >= 0, y >= 0, z = x, z' = y b(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1 encArg(z) -{ 0 }-> b(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> a(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_a(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_b(z, z') -{ 0 }-> b(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_b(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_c(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 ---------------------------------------- (35) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: a(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 a(z, z') -{ 1 }-> b(a(1 + 0, z' - 3), 0) :|: z >= 0, z' - 3 >= 0 a(z, z') -{ 2 }-> b(a(1 + (1 + a(1 + z, a(0, 0))), z' - 3), 0) :|: z >= 0, z' - 3 >= 0 a(z, z') -{ 1 }-> b(a(1 + (1 + 0 + z), z' - 3), 0) :|: z >= 0, z' - 3 >= 0 a(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 b(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 b(z, z') -{ 2 }-> 1 + a(1 + z', b(a(1 + b(0, 0), z - 3), 0)) :|: z - 3 >= 0, z' >= 0 b(z, z') -{ 2 }-> 1 + a(1 + z', 0) :|: z >= 0, z' >= 0 b(z, z') -{ 1 }-> 1 + a(1 + z', 0) :|: z >= 0, z' >= 0 b(z, z') -{ 1 }-> 1 + a(1 + z', 1 + 0 + z) :|: z >= 0, z' >= 0 b(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0 encArg(z) -{ 0 }-> b(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> a(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_a(z, z') -{ 0 }-> a(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_a(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_b(z, z') -{ 0 }-> b(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_b(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_c(z) -{ 0 }-> 0 :|: z >= 0 encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 ---------------------------------------- (37) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) encode_b(z0, z1) -> b(encArg(z0), encArg(z1)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(0) -> c2 ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_B(z0, z1) -> c5(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_C(z0) -> c6(ENCARG(z0)) ENCODE_A(z0, z1) -> c7(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_0 -> c8 A(z0, z1) -> c9 A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c12 B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) S tuples: A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) K tuples:none Defined Rule Symbols: b_2, a_2, encArg_1, encode_b_2, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_B_2, ENCODE_C_1, ENCODE_A_2, ENCODE_0, A_2, B_2 Compound Symbols: c1_1, c2, c3_3, c4_3, c5_3, c6_1, c7_3, c8, c9, c10, c11_3, c12, c13_2 ---------------------------------------- (39) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_C(z0) -> c6(ENCARG(z0)) Removed 4 trailing nodes: ENCODE_0 -> c8 A(z0, z1) -> c9 B(z0, z1) -> c12 ENCARG(0) -> c2 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) encode_b(z0, z1) -> b(encArg(z0), encArg(z1)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_B(z0, z1) -> c5(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_A(z0, z1) -> c7(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) S tuples: A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) K tuples:none Defined Rule Symbols: b_2, a_2, encArg_1, encode_b_2, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_B_2, ENCODE_A_2, A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c5_3, c7_3, c10, c11_3, c13_2 ---------------------------------------- (41) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) encode_b(z0, z1) -> b(encArg(z0), encArg(z1)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_B(z0, z1) -> c2(ENCARG(z0)) ENCODE_B(z0, z1) -> c2(ENCARG(z1)) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(ENCARG(z0)) ENCODE_A(z0, z1) -> c2(ENCARG(z1)) S tuples: A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) K tuples:none Defined Rule Symbols: b_2, a_2, encArg_1, encode_b_2, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, A_2, B_2, ENCODE_B_2, ENCODE_A_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c11_3, c13_2, c2_1 ---------------------------------------- (43) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 4 leading nodes: ENCODE_B(z0, z1) -> c2(ENCARG(z0)) ENCODE_B(z0, z1) -> c2(ENCARG(z1)) ENCODE_A(z0, z1) -> c2(ENCARG(z0)) ENCODE_A(z0, z1) -> c2(ENCARG(z1)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) encode_b(z0, z1) -> b(encArg(z0), encArg(z1)) encode_c(z0) -> c(encArg(z0)) encode_a(z0, z1) -> a(encArg(z0), encArg(z1)) encode_0 -> 0 a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) S tuples: A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) K tuples:none Defined Rule Symbols: b_2, a_2, encArg_1, encode_b_2, encode_c_1, encode_a_2, encode_0 Defined Pair Symbols: ENCARG_1, A_2, B_2, ENCODE_B_2, ENCODE_A_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c11_3, c13_2, c2_1 ---------------------------------------- (45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_b(z0, z1) -> b(encArg(z0), encArg(z1)) 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(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) S tuples: A(z0, z1) -> c10 A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, B_2, ENCODE_B_2, ENCODE_A_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c11_3, c13_2, c2_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(z0, c(c_b(c_a(0, z1), 0))) -> c11(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0)) by A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) S tuples: A(z0, z1) -> c10 B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, B_2, ENCODE_B_2, ENCODE_A_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c13_2, c2_1, c11_3 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(z0, z1) -> c13(A(c(z1), a(0, z0)), A(0, z0)) by B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) S tuples: A(z0, z1) -> c10 A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c11_3, c13_2 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) by A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) S tuples: A(z0, z1) -> c10 A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c11_3, c13_2, c11_1 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) by A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) S tuples: A(z0, z1) -> c10 A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c11_3, c13_2, c11_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) by A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) S tuples: A(z0, z1) -> c10 A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c11_3, c13_2, c11_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) by A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(c_b(0, x0)), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(c_b(0, x0)), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) S tuples: A(z0, z1) -> c10 A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c11_3, c13_2, c11_1, c11_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) by A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c_a(c(c(a(c(x0), a(0, 0)))), z1), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, z1), 0))) -> c11(B(c(c(a(c(x0), a(0, 0)))), 0), A(c(b(0, x0)), z1), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(x0), a(0, 0)))))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) S tuples: A(z0, z1) -> c10 B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c13_2, c11_3, c11_1, c11_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) by B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(c(z0)) -> c1(ENCARG(z0)) ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) S tuples: A(z0, z1) -> c10 B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c1_1, c3_3, c4_3, c10, c2_1, c13_2, c11_3, c11_1, c11_2, c13_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(c(z0)) -> c1(ENCARG(z0)) by ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_b(y0, y1))) -> c1(ENCARG(cons_b(y0, y1))) ENCARG(c(cons_a(y0, y1))) -> c1(ENCARG(cons_a(y0, y1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) ENCARG(c(cons_b(y0, y1))) -> c1(ENCARG(cons_b(y0, y1))) ENCARG(c(cons_a(y0, y1))) -> c1(ENCARG(cons_a(y0, y1))) S tuples: A(z0, z1) -> c10 B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c3_3, c4_3, c10, c2_1, c13_2, c11_3, c11_1, c11_2, c13_1, c1_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ENCARG(c(c(y0))) -> c1(ENCARG(c(y0))) by ENCARG(c(c(c(y0)))) -> c1(ENCARG(c(c(y0)))) ENCARG(c(c(cons_b(y0, y1)))) -> c1(ENCARG(c(cons_b(y0, y1)))) ENCARG(c(c(cons_a(y0, y1)))) -> c1(ENCARG(c(cons_a(y0, y1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: encArg(c(z0)) -> c(encArg(z0)) encArg(0) -> 0 encArg(cons_b(z0, z1)) -> b(encArg(z0), encArg(z1)) encArg(cons_a(z0, z1)) -> a(encArg(z0), encArg(z1)) b(z0, z1) -> c_b(z0, z1) b(z0, z1) -> c(a(c(z1), a(0, z0))) a(z0, z1) -> c_a(z0, z1) a(z0, z1) -> z0 a(z0, c(c_b(c_a(0, z1), 0))) -> b(a(c(b(0, z0)), z1), 0) Tuples: ENCARG(cons_b(z0, z1)) -> c3(B(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_a(z0, z1)) -> c4(A(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) A(z0, z1) -> c10 ENCODE_B(z0, z1) -> c2(B(encArg(z0), encArg(z1))) ENCODE_A(z0, z1) -> c2(A(encArg(z0), encArg(z1))) B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) ENCARG(c(cons_b(y0, y1))) -> c1(ENCARG(cons_b(y0, y1))) ENCARG(c(cons_a(y0, y1))) -> c1(ENCARG(cons_a(y0, y1))) ENCARG(c(c(c(y0)))) -> c1(ENCARG(c(c(y0)))) ENCARG(c(c(cons_b(y0, y1)))) -> c1(ENCARG(c(cons_b(y0, y1)))) ENCARG(c(c(cons_a(y0, y1)))) -> c1(ENCARG(c(cons_a(y0, y1)))) S tuples: A(z0, z1) -> c10 B(z1, x1) -> c13(A(c(x1), c_a(0, z1)), A(0, z1)) B(z1, x1) -> c13(A(c(x1), 0), A(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c_b(0, z1)), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c_a(c(c(a(c(z1), a(0, 0)))), x1), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c_b(0, z1)), 0), A(c(b(0, z1)), x1), B(0, z1)) A(z1, c(c_b(c_a(0, x1), 0))) -> c11(B(c(c(a(c(z1), a(0, 0)))), 0), A(c(b(0, z1)), x1), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c_b(a(c(b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(c(a(c(0), a(0, a(c(b(0, c(b(0, x0)))), x1)))), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c_a(c(b(0, c(b(0, x0)))), z1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))) -> c11(B(b(c(b(0, c(b(0, x0)))), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, z1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), 0))) -> c11(B(b(b(a(c(b(0, c(b(0, c(b(0, x0)))))), z1), 0), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c_b(0, c(b(0, x0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(c(a(c(c(b(0, x0))), a(0, 0)))), x1), 0), 0), A(c(b(0, x0)), c(c_b(c_a(0, x1), 0))), B(0, x0)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c_b(0, z1)))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(z1, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(B(b(a(c(b(0, c(c(a(c(z1), a(0, 0)))))), x1), 0), 0), A(c(b(0, z1)), c(c_b(c_a(0, x1), 0))), B(0, z1)) A(x0, c(c_b(c_a(0, c(c_b(c_a(0, x1), 0))), 0))) -> c11(A(c(b(0, x0)), c(c_b(c_a(0, x1), 0)))) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c_a(c(x0), a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(c(x0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), c_a(0, 0)))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) A(x0, c(c_b(c_a(0, x1), 0))) -> c11(B(a(c(c(a(c(x0), 0))), x1), 0), A(c(b(0, x0)), x1), B(0, x0)) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c_b(a(c(b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), c(a(c(0), a(0, a(c(b(0, 0)), x0))))), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c_a(c(b(0, 0)), z1), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, z1), 0)), x1) -> c13(A(c(x1), b(c(b(0, 0)), 0)), A(0, c(c_b(c_a(0, z1), 0)))) B(c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)), x1) -> c13(A(c(x1), b(b(a(c(b(0, c(b(0, 0)))), z1), 0), 0)), A(0, c(c_b(c_a(0, c(c_b(c_a(0, z1), 0))), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c_b(0, 0)), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(c(x1), b(a(c(c(a(c(0), a(0, 0)))), x0), 0)), A(0, c(c_b(c_a(0, x0), 0)))) B(c(c_b(c_a(0, x0), 0)), x1) -> c13(A(0, c(c_b(c_a(0, x0), 0)))) K tuples:none Defined Rule Symbols: encArg_1, b_2, a_2 Defined Pair Symbols: ENCARG_1, A_2, ENCODE_B_2, ENCODE_A_2, B_2 Compound Symbols: c3_3, c4_3, c10, c2_1, c13_2, c11_3, c11_1, c11_2, c13_1, c1_1