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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | 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
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