/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 48 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 154 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 (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence twice(s(x1)) ->^+ p(p(s(s(s(p(p(s(s(s(twice(x1))))))))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0,0,0,0,0,0,0]. The pumping substitution is [x1 / s(x1)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: sq(0(x1)) -> p(s(p(s(p(p(p(p(s(s(s(s(0(p(s(p(s(x1))))))))))))))))) sq(s(x1)) -> s(p(s(p(s(p(p(s(s(twice(p(s(p(s(p(p(p(s(s(s(sq(p(p(p(p(p(p(s(s(s(s(s(s(x1))))))))))))))))))))))))))))))))) twice(0(x1)) -> p(p(p(p(s(s(p(s(s(s(0(p(p(p(s(s(s(p(p(s(s(p(s(p(s(p(s(x1))))))))))))))))))))))))))) twice(s(x1)) -> p(p(s(s(s(p(p(s(s(s(twice(p(s(p(s(x1))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(s(s(s(x1)))))))))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_sq(x_1)) -> sq(encArg(x_1)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_sq(x_1) -> sq(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_twice(x_1) -> twice(encArg(x_1)) Rewrite Strategy: FULL