| import OAI.Combinatorics.PerfectMatching.Simplex |
| import Mathlib.Geometry.Convex.ConvexSpace.CompactSpaceStdSimplex |
| import Mathlib.Geometry.Convex.ConvexSpace.Barycenter |
| import Mathlib.Analysis.Convex.Contractible |
| import Mathlib.Analysis.Convex.Topology |
| import Mathlib.Topology.MetricSpace.Basic |
| import Mathlib.Topology.Algebra.Module.LocallyConvex |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Convexity |
|
|
| namespace Problem340.RealSimplex |
|
|
| variable {Index : Type*} [Fintype Index] |
| |
| instance coordinatesFun : CoeFun (StdSimplex ℝ Index) (fun _ => Index → ℝ) := |
| ⟨fun point => point.weights⟩ |
| |
| def coordinates (point : StdSimplex ℝ Index) : Index → ℝ := point.weights |
| |
| omit [Fintype Index] in |
| @[simp] lemma single_apply [DecidableEq Index] (vertex index : Index) : |
| StdSimplex.single (R := ℝ) vertex index = if vertex = index then 1 else 0 := by |
| change (StdSimplex.single vertex).weights index = _ |
| simp [Finsupp.single_apply] |
| |
| @[simp] lemma barycenter_apply [Nonempty Index] (index : Index) : |
| (StdSimplex.barycenter : StdSimplex ℝ Index) index = (Fintype.card Index : ℝ)⁻¹ := |
| StdSimplex.weights_barycenter_apply index |
| |
| lemma map_apply {Target : Type*} [Fintype Target] [DecidableEq Target] |
| (mapping : Index → Target) (point : StdSimplex ℝ Index) (index : Target) : |
| point.map mapping index = |
| ∑ source ∈ Finset.univ.filter (fun source => mapping source = index), point source := by |
| classical |
| change (point.map mapping).weights index = _ |
| rw [StdSimplex.weights_map, Finsupp.mapDomain_apply, |
| Finsupp.sum_fintype _ _ (by simp), Finset.sum_filter] |
| simp [Finsupp.single_apply] |
| |
| lemma coordinates_injective : Function.Injective (coordinates (Index := Index)) := |
| (StdSimplex.isEmbedding_toFun_comp_weights ℝ Index).injective |
| |
| def coordinateSet (Index : Type*) [Fintype Index] : Set (Index → ℝ) := |
| {weights | (∀ index, 0 ≤ weights index) ∧ ∑ index, weights index = 1} |
| |
| lemma mem_coordinateSet (point : StdSimplex ℝ Index) : |
| coordinates point ∈ coordinateSet Index := |
| ⟨point.weights_nonneg, point.total_of_fintype⟩ |
| |
| lemma convex_coordinateSet : Convex ℝ (coordinateSet Index) := |
| MatchingEntropy.convex_probabilitySimplex Index |
| |
| def coordinateHomeomorph : StdSimplex ℝ Index ≃ₜ coordinateSet Index := |
| (StdSimplex.isEmbedding_toFun_comp_weights ℝ Index).toHomeomorph.trans |
| (Homeomorph.setCongr (by |
| rw [StdSimplex.range_toFun_comp_weights] |
| ext weights |
| simp [coordinateSet])) |
| |
| @[simp] lemma coordinateHomeomorph_apply (point : StdSimplex ℝ Index) : |
| (coordinateHomeomorph point : Index → ℝ) = coordinates point := rfl |
| |
| @[simp] lemma coordinateHomeomorph_symm_apply (point : coordinateSet Index) : |
| coordinates (coordinateHomeomorph.symm point) = point := |
| congrArg Subtype.val (coordinateHomeomorph.apply_symm_apply point) |
| |
| instance metricSpace : MetricSpace (StdSimplex ℝ Index) := |
| (StdSimplex.isEmbedding_toFun_comp_weights ℝ Index).comapMetricSpace _ |
| |
| @[simp] lemma coordinateHomeomorph_symm_apply_at (point : coordinateSet Index) (index : Index) : |
| coordinateHomeomorph.symm point index = point.val index := |
| congrFun (coordinateHomeomorph_symm_apply point) index |
| |
| lemma contractible [Nonempty Index] : ContractibleSpace (StdSimplex ℝ Index) := by |
| have : ContractibleSpace (coordinateSet Index) := |
| convex_coordinateSet.contractibleSpace |
| ⟨coordinates (StdSimplex.single (Classical.arbitrary Index)), mem_coordinateSet _⟩ |
| exact (coordinateHomeomorph (Index := Index)).contractibleSpace |
| |
| lemma locallyPathConnected : LocallyPathConnectedSpace (StdSimplex ℝ Index) := by |
| have : LocallyPathConnectedSpace (coordinateSet Index) := |
| convex_coordinateSet.locallyPathConnectedSpace |
| exact (coordinateHomeomorph (Index := Index)).symm.locallyPathConnectedSpace |
| |
| end Problem340.RealSimplex |
| |
| end |
| |
| end OAI |
| |
