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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 179 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 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 (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) The (relative) TRS S consists of the following rules: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) The (relative) TRS S consists of the following rules: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) The (relative) TRS S consists of the following rules: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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 not(or(x, y)) ->^+ and(not(not(not(x))), not(not(not(y)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. The pumping substitution is [x / or(x, y)]. 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: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) The (relative) TRS S consists of the following rules: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) 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: not(not(x)) -> x not(or(x, y)) -> and(not(not(not(x))), not(not(not(y)))) not(and(x, y)) -> or(not(not(not(x))), not(not(not(y)))) The (relative) TRS S consists of the following rules: encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) encArg(and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_not(x_1)) -> not(encArg(x_1)) encode_not(x_1) -> not(encArg(x_1)) encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) Rewrite Strategy: FULL