Dev/MonadicDependence/WeaklySparseMonDepIsNowhereDense/Contract.lean
| 1 | import Catalog.Sparsity.Preliminaries.Contract |
| 2 | import Dev.MonadicDependence.WeaklySparse.Contract |
| 3 | import Dev.MonadicDependence.MonadicDependence.Contract |
| 4 | import Dev.MonadicDependence.NowhereDense.Contract |
| 5 | |
| 6 | namespace Dev.MonadicDependence.WeaklySparseMonDepIsNowhereDense |
| 7 | |
| 8 | open Catalog.Sparsity.Preliminaries |
| 9 | open Dev.MonadicDependence.WeaklySparse |
| 10 | open Dev.MonadicDependence.MonadicDependence |
| 11 | open Dev.MonadicDependence.NowhereDense |
| 12 | |
| 13 | /-- Mählmann Lemma 13.7. If a graph class `C` is weakly sparse |
| 14 | (`Dev.MonadicDependence.WeaklySparse.IsWeaklySparse`) and monadically |
| 15 | dependent (`Dev.MonadicDependence.MonadicDependence.IsMonadicallyDependent`), |
| 16 | then it is nowhere dense in the local sense of |
| 17 | `Dev.MonadicDependence.NowhereDense.IsNowhereDense`. -/ |
| 18 | axiom weaklySparseMonDepIsNowhereDense (C : GraphClass) : |
| 19 | IsWeaklySparse C → IsMonadicallyDependent C → IsNowhereDense C |
| 20 | |
| 21 | end Dev.MonadicDependence.WeaklySparseMonDepIsNowhereDense |
| 22 |