/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), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 21 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 3 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(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(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(x1))) The (relative) TRS S consists of the following rules: encArg(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(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, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(x1))) The (relative) TRS S consists of the following rules: encArg(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(x1))) The (relative) TRS S consists of the following rules: encArg(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence 1(q9(x1)) ->^+ q9(1(x1)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x1 / q9(x1)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) 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, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(x1))) The (relative) TRS S consists of the following rules: encArg(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: 0(q0(0(x1))) -> 0(0(q0(x1))) 0(q0(1(x1))) -> 0(1(q0(x1))) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 0(q1(0(x1))) -> 0(0(q2(x1))) 0(q1(1(x1))) -> 0(1(q2(x1))) 1(q2(0(x1))) -> 1(0(q2(x1))) 1(q2(1(x1))) -> 1(1(q2(x1))) 0(q2(x1)) -> q3(1(x1)) 1(q3(x1)) -> q3(1(x1)) 0(q3(x1)) -> q4(0(x1)) 1(q4(x1)) -> q4(1(x1)) 0(q4(0(x1))) -> 1(0(q5(x1))) 0(q4(1(x1))) -> 1(1(q5(x1))) 1(q5(0(x1))) -> 0(0(q1(x1))) 1(q5(1(x1))) -> 0(1(q1(x1))) 0(q5(x1)) -> q6(0(x1)) 1(q6(x1)) -> q6(1(x1)) 1(q7(0(x1))) -> 0(0(q8(x1))) 1(q7(1(x1))) -> 0(1(q8(x1))) 0(q8(x1)) -> 0(q0(x1)) 1(q8(0(x1))) -> 1(0(q8(x1))) 1(q8(1(x1))) -> 1(1(q8(x1))) 0(q6(x1)) -> q9(0(x1)) 0(q9(0(x1))) -> 1(0(q7(x1))) 0(q9(1(x1))) -> 1(1(q7(x1))) 1(q9(x1)) -> q9(1(x1)) h(q0(x1)) -> h(0(q0(x1))) q0(h(x1)) -> q0(0(h(x1))) h(q1(x1)) -> h(0(q1(x1))) q1(h(x1)) -> q1(0(h(x1))) h(q2(x1)) -> h(0(q2(x1))) q2(h(x1)) -> q2(0(h(x1))) h(q3(x1)) -> h(0(q3(x1))) q3(h(x1)) -> q3(0(h(x1))) h(q4(x1)) -> h(0(q4(x1))) q4(h(x1)) -> q4(0(h(x1))) h(q5(x1)) -> h(0(q5(x1))) q5(h(x1)) -> q5(0(h(x1))) h(q6(x1)) -> h(0(q6(x1))) q6(h(x1)) -> q6(0(h(x1))) The (relative) TRS S consists of the following rules: encArg(q7(x_1)) -> q7(encArg(x_1)) encArg(q8(x_1)) -> q8(encArg(x_1)) encArg(q9(x_1)) -> q9(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_q0(x_1)) -> q0(encArg(x_1)) encArg(cons_q1(x_1)) -> q1(encArg(x_1)) encArg(cons_q2(x_1)) -> q2(encArg(x_1)) encArg(cons_q3(x_1)) -> q3(encArg(x_1)) encArg(cons_q4(x_1)) -> q4(encArg(x_1)) encArg(cons_q5(x_1)) -> q5(encArg(x_1)) encArg(cons_q6(x_1)) -> q6(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_q0(x_1) -> q0(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_q1(x_1) -> q1(encArg(x_1)) encode_q2(x_1) -> q2(encArg(x_1)) encode_q3(x_1) -> q3(encArg(x_1)) encode_q4(x_1) -> q4(encArg(x_1)) encode_q5(x_1) -> q5(encArg(x_1)) encode_q6(x_1) -> q6(encArg(x_1)) encode_q7(x_1) -> q7(encArg(x_1)) encode_q8(x_1) -> q8(encArg(x_1)) encode_q9(x_1) -> q9(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: INNERMOST