/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 72 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) CpxTrsMatchBoundsProof [FINISHED, 155 ms] (8) BOUNDS(1, n^1) (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 3 ms] (10) TRS for Loop Detection (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. The certificate found is represented by the following graph. 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(425,364,[0_1|3]), (425,368,[0_1|3]), (426,427,[0_1|3]), (427,428,[1_1|3]), (428,429,[2_1|3]), (429,430,[2_1|3]), (430,134,[4_1|3]), (430,138,[4_1|3]), (430,142,[4_1|3]), (430,146,[4_1|3]), (430,150,[4_1|3]), (430,155,[4_1|3]), (430,163,[4_1|3]), (430,168,[4_1|3]), (430,173,[4_1|3]), (430,178,[4_1|3]), (430,182,[4_1|3]), (430,202,[4_1|3]), (430,277,[4_1|3]), (430,282,[4_1|3]), (430,287,[4_1|3]), (430,292,[4_1|3]), (430,297,[4_1|3]), (430,316,[4_1|3]), (430,345,[4_1|3]), (430,350,[4_1|3]), (430,359,[4_1|3]), (430,363,[4_1|3]), (430,367,[4_1|3]), (430,135,[4_1|3]), (430,139,[4_1|3]), (430,143,[4_1|3]), (430,372,[4_1|3]), (430,376,[4_1|3]), (430,147,[4_1|3]), (430,151,[4_1|3]), (430,156,[4_1|3]), (430,164,[4_1|3]), (430,169,[4_1|3]), (430,174,[4_1|3]), (430,183,[4_1|3]), (430,288,[4_1|3]), (430,293,[4_1|3]), (430,298,[4_1|3]), (430,219,[4_1|3]), (430,351,[4_1|3]), (430,360,[4_1|3]), (430,364,[4_1|3]), (430,368,[4_1|3]), (431,432,[0_1|3]), (432,433,[2_1|3]), (433,434,[4_1|3]), (434,435,[1_1|3]), (435,134,[4_1|3]), (435,138,[4_1|3]), (435,142,[4_1|3]), (435,146,[4_1|3]), (435,150,[4_1|3]), (435,155,[4_1|3]), (435,163,[4_1|3]), (435,168,[4_1|3]), (435,173,[4_1|3]), (435,178,[4_1|3]), (435,182,[4_1|3]), (435,202,[4_1|3]), (435,277,[4_1|3]), (435,282,[4_1|3]), (435,287,[4_1|3]), (435,292,[4_1|3]), (435,297,[4_1|3]), (435,316,[4_1|3]), (435,345,[4_1|3]), (435,350,[4_1|3]), (435,359,[4_1|3]), (435,363,[4_1|3]), (435,367,[4_1|3]), (435,135,[4_1|3]), (435,139,[4_1|3]), (435,143,[4_1|3]), (435,372,[4_1|3]), (435,376,[4_1|3]), (435,147,[4_1|3]), (435,151,[4_1|3]), (435,156,[4_1|3]), (435,164,[4_1|3]), (435,169,[4_1|3]), (435,174,[4_1|3]), (435,183,[4_1|3]), (435,288,[4_1|3]), (435,293,[4_1|3]), (435,298,[4_1|3]), (435,219,[4_1|3]), (435,351,[4_1|3]), (435,360,[4_1|3]), (435,364,[4_1|3]), (435,368,[4_1|3]), (436,437,[0_1|3]), (437,438,[3_1|3]), (438,439,[1_1|3]), (439,307,[4_1|3]), (439,180,[4_1|3]), (440,441,[3_1|3]), (441,442,[0_1|3]), (442,443,[1_1|3]), (443,307,[2_1|3]), (443,180,[2_1|3]), (444,445,[0_1|3]), (445,446,[1_1|3]), (446,447,[3_1|3]), (447,448,[2_1|3]), (448,307,[5_1|3]), (448,180,[5_1|3]), (449,450,[1_1|3]), (450,451,[1_1|3]), (451,452,[4_1|3]), (452,453,[5_1|3]), (453,312,[0_1|3]), (453,331,[0_1|3]), (453,318,[0_1|3]), (454,455,[0_1|3]), (455,456,[3_1|3]), (456,457,[1_1|3]), (457,246,[4_1|3]), (458,459,[0_1|2]), (459,460,[2_1|2]), (460,461,[1_1|2]), (461,128,[4_1|2]), (461,192,[4_1|2]), (461,197,[4_1|2]), (461,212,[4_1|2]), (461,354,[4_1|2]), (462,463,[2_1|2]), (463,464,[4_1|2]), (464,465,[2_1|2]), (465,466,[5_1|2]), (466,128,[0_1|2]), (466,192,[0_1|2, 1_1|2]), (466,197,[0_1|2, 1_1|2]), (466,212,[0_1|2, 1_1|2]), (466,354,[0_1|2, 1_1|2]), (466,134,[0_1|2]), (466,138,[0_1|2]), (466,142,[0_1|2]), (466,146,[0_1|2]), (466,150,[0_1|2]), (466,155,[0_1|2]), (466,159,[2_1|2]), (466,163,[0_1|2]), (466,168,[0_1|2]), (466,173,[0_1|2]), (466,178,[0_1|2]), (466,182,[0_1|2]), (466,187,[3_1|2]), (466,350,[0_1|2]), (466,202,[0_1|2]), (466,207,[2_1|2]), (466,217,[4_1|2]), (466,222,[5_1|2]), (466,227,[5_1|2]), (466,232,[4_1|2]), (466,237,[5_1|2]), (466,242,[5_1|2]), (466,247,[3_1|2]), (466,252,[3_1|2]), (466,257,[3_1|2]), (466,262,[5_1|2]), (466,267,[5_1|2]), (466,272,[5_1|2]), (466,277,[0_1|2]), (466,282,[0_1|2]), (466,287,[0_1|2]), (466,292,[0_1|2]), (466,297,[0_1|2]), (466,301,[5_1|2]), (466,359,[0_1|3]), (466,363,[0_1|3]), (466,367,[0_1|3]), (467,468,[4_1|3]), (468,469,[3_1|3]), (469,470,[2_1|3]), (470,471,[0_1|3]), (470,359,[0_1|3]), (470,363,[0_1|3]), (471,134,[0_1|3]), (471,138,[0_1|3]), (471,142,[0_1|3]), (471,146,[0_1|3]), (471,150,[0_1|3]), (471,155,[0_1|3]), (471,163,[0_1|3]), (471,168,[0_1|3]), (471,173,[0_1|3]), (471,178,[0_1|3]), (471,182,[0_1|3]), (471,202,[0_1|3]), (471,277,[0_1|3]), (471,282,[0_1|3]), (471,287,[0_1|3]), (471,292,[0_1|3]), (471,297,[0_1|3]), (471,316,[0_1|3]), (471,345,[0_1|3]), (471,350,[0_1|3]), (471,359,[0_1|3]), (471,363,[0_1|3]), (471,367,[0_1|3]), (471,135,[0_1|3]), (471,139,[0_1|3]), (471,143,[0_1|3]), (471,372,[0_1|3]), (471,376,[0_1|3]), (471,147,[0_1|3]), (471,151,[0_1|3]), (471,156,[0_1|3]), (471,164,[0_1|3]), (471,169,[0_1|3]), (471,174,[0_1|3]), (471,183,[0_1|3]), (471,288,[0_1|3]), (471,293,[0_1|3]), (471,298,[0_1|3]), (471,351,[0_1|3]), (471,360,[0_1|3]), (471,364,[0_1|3]), (471,368,[0_1|3]), (472,473,[2_1|3]), (473,474,[4_1|3]), (474,475,[2_1|3]), (475,476,[5_1|3]), (476,312,[0_1|3]), (476,331,[0_1|3]), (476,318,[0_1|3]), (477,478,[1_1|3]), (478,479,[4_1|3]), (479,480,[5_1|3]), (480,481,[0_1|3]), (481,312,[2_1|3]), (481,331,[2_1|3]), (481,318,[2_1|3]), (482,483,[2_1|3]), (483,484,[5_1|3]), (484,485,[1_1|3]), (485,486,[4_1|3]), (486,312,[0_1|3]), (486,331,[0_1|3]), (486,318,[0_1|3]), (487,488,[3_1|3]), (488,489,[0_1|3]), (489,490,[5_1|3]), (490,491,[1_1|3]), (491,312,[4_1|3]), (491,331,[4_1|3]), (491,318,[4_1|3]), (492,493,[5_1|3]), (493,494,[0_1|3]), (494,495,[5_1|3]), (495,496,[1_1|3]), (496,312,[4_1|3]), (496,331,[4_1|3]), (496,318,[4_1|3]), (497,498,[0_1|2]), (498,499,[3_1|2]), (499,500,[1_1|2]), (500,128,[4_1|2]), (500,134,[4_1|2]), (500,138,[4_1|2]), (500,142,[4_1|2]), (500,146,[4_1|2]), (500,150,[4_1|2]), (500,155,[4_1|2]), (500,163,[4_1|2]), (500,168,[4_1|2]), (500,173,[4_1|2]), (500,178,[4_1|2]), (500,182,[4_1|2]), (500,202,[4_1|2]), (500,277,[4_1|2]), (500,282,[4_1|2]), (500,287,[4_1|2]), (500,292,[4_1|2]), (500,297,[4_1|2]), (500,316,[4_1|2]), (500,345,[4_1|2]), (500,350,[4_1|2]), (500,359,[4_1|2]), (500,363,[4_1|2]), (500,367,[4_1|2]), (501,502,[0_1|4]), (502,503,[2_1|4]), (503,504,[1_1|4]), (504,420,[2_1|4]), (504,428,[2_1|4]), (504,446,[2_1|4]), (505,506,[0_1|4]), (506,507,[3_1|4]), (507,508,[1_1|4]), (508,420,[4_1|4]), (508,428,[4_1|4]), (508,446,[4_1|4]), (509,510,[0_1|4]), (510,511,[3_1|4]), (511,512,[1_1|4]), (512,513,[2_1|4]), (513,443,[2_1|4]), (514,515,[0_1|3]), (515,516,[1_1|3]), (516,517,[2_1|3]), (517,219,[4_1|3]), (518,519,[1_1|3]), (519,520,[2_1|3]), (520,521,[0_1|3]), (521,219,[0_1|3]), (522,523,[0_1|3]), (523,524,[1_1|3]), (524,525,[2_1|3]), (525,526,[2_1|3]), (526,219,[4_1|3]), (527,528,[0_1|3]), (528,529,[2_1|3]), (529,530,[4_1|3]), (530,531,[1_1|3]), (531,219,[4_1|3]), (532,533,[0_1|3]), (533,534,[2_1|3]), (534,535,[1_1|3]), (535,157,[4_1|3]), (535,165,[4_1|3]), (535,184,[4_1|3]), (536,537,[0_1|3]), (537,538,[3_1|3]), (538,539,[1_1|3]), (539,540,[2_1|3]), (540,157,[2_1|3]), (540,165,[2_1|3]), (540,184,[2_1|3]), (541,542,[0_1|4]), (542,543,[2_1|4]), (543,544,[1_1|4]), (544,413,[2_1|4]), (545,546,[0_1|4]), (546,547,[3_1|4]), (547,548,[1_1|4]), (548,413,[4_1|4]), (549,550,[2_1|4]), (550,551,[0_1|4]), (551,552,[3_1|4]), (552,553,[1_1|4]), (553,372,[4_1|4]), (553,376,[4_1|4]), (553,360,[4_1|4]), (553,364,[4_1|4]), (553,368,[4_1|4]), (553,380,[4_1|4]), (553,373,[4_1|4]), (553,377,[4_1|4]), (554,555,[0_1|4]), (555,556,[2_1|4]), (556,557,[1_1|4]), (557,516,[2_1|4]), (557,524,[2_1|4]), (558,559,[0_1|4]), (559,560,[3_1|4]), (560,561,[1_1|4]), (561,516,[4_1|4]), (561,524,[4_1|4])}" ---------------------------------------- (8) BOUNDS(1, n^1) ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (11) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 0(1(5(4(x1)))) ->^+ 1(2(4(2(5(0(x1)))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0]. The pumping substitution is [x1 / 1(5(4(x1)))]. The result substitution is [ ]. ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) The (relative) TRS S consists of the following rules: encArg(1(x_1)) -> 1(encArg(x_1)) encArg(2(x_1)) -> 2(encArg(x_1)) encArg(4(x_1)) -> 4(encArg(x_1)) encArg(5(x_1)) -> 5(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_3(x_1)) -> 3(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_2(x_1) -> 2(encArg(x_1)) encode_3(x_1) -> 3(encArg(x_1)) encode_4(x_1) -> 4(encArg(x_1)) encode_5(x_1) -> 5(encArg(x_1)) Rewrite Strategy: INNERMOST