| import OAI.Analysis.DiamondDistortion.PathBounds |
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| namespace OAI |
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| universe u v |
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| noncomputable section |
| open scoped BigOperators Classical Topology |
| open Set |
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| namespace DiamondDistortion |
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| namespace CoordinateCompletion |
| variable {ι : Type*} |
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|
| /-- A specified coordinate norm, with individually bounded coordinate evaluations. |
| No bound on the norms of coordinate projections is imposed. -/ |
| structure Data (ι : Type*) where |
| seminorm : Seminorm ℝ (ι →₀ ℝ) |
| coordinate_bound : ∀ i,∃ C : ℝ,∀ v,|v i| ≤ C*seminorm v |
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| namespace Data |
| variable (A : Data ι) |
| lemma separates (v : ι →₀ ℝ) (hv : A.seminorm v=0) : v=0 := by |
| ext i |
| obtain ⟨C,hC⟩ := A.coordinate_bound i |
| have hh := hC v |
| rw [hv,mul_zero] at hh |
| exact abs_nonpos_iff.mp hh |
|
|
| def Finite (_A : Data ι) := ι →₀ ℝ |
| instance : AddCommGroup A.Finite := inferInstanceAs (AddCommGroup (ι →₀ ℝ)) |
| instance : Module ℝ A.Finite := inferInstanceAs (Module ℝ (ι →₀ ℝ)) |
| instance : NormedAddCommGroup A.Finite := AddGroupNorm.toNormedAddCommGroup |
| { A.seminorm.toAddGroupSeminorm with eq_zero_of_map_eq_zero' := A.separates } |
| instance : NormedSpace ℝ A.Finite where |
| norm_smul_le t v := le_of_eq (A.seminorm.smul' t v) |
| abbrev Space := UniformSpace.Completion A.Finite |
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|
| def finite (v : ι →₀ ℝ) : A.Finite := v |
| def array (v : A.Finite) : ι →₀ ℝ := v |
| def embedding : (ι →₀ ℝ) →ₗ[ℝ] A.Space := |
| (UniformSpace.Completion.toComplL (S := ℝ) (α := A.Finite)).toLinearMap |
| @[simp] lemma norm_embedding (v : ι →₀ ℝ) : ‖A.embedding v‖=A.seminorm v := |
| UniformSpace.Completion.norm_coe _ |
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| def coordinateFinite (i : ι) : A.Finite →L[ℝ] ℝ := |
| (show A.Finite →ₗ[ℝ] ℝ from Finsupp.lapply i).mkContinuous |
| (A.coordinate_bound i).choose (by |
| intro v |
| change ‖(show ι →₀ ℝ from v) i‖ ≤ (A.coordinate_bound i).choose*A.seminorm v |
| simpa only [Real.norm_eq_abs] using (A.coordinate_bound i).choose_spec v) |
| def coordinate (i : ι) : A.Space →L[ℝ] ℝ := |
| (A.coordinateFinite i).extend UniformSpace.Completion.toComplL |
| @[simp] lemma coordinate_coe (i : ι) (v : A.Finite) : A.coordinate i (v : A.Space)=A.array v i := |
| ContinuousLinearMap.extend_eq _ UniformSpace.Completion.denseRange_coe |
| (UniformSpace.Completion.isUniformInducing_coe _) v |
| @[simp] lemma coordinate_embedding (i : ι) (v : ι →₀ ℝ) : A.coordinate i (A.embedding v)=v i := |
| A.coordinate_coe i (A.finite v) |
| def unit (i : ι) : A.Space := A.embedding (Finsupp.single i 1) |
| @[simp] lemma coordinate_unit (i j : ι) : A.coordinate i (A.unit j)=if j=i then 1 else 0 := by |
| simp [unit,Finsupp.single_apply] |
|
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| def projection (H : Finset ι) : A.Space →L[ℝ] A.Space := |
| ∑ i ∈ H,(A.coordinate i).smulRight (A.unit i) |
| @[simp] lemma projection_apply (H : Finset ι) (x : A.Space) : |
| A.projection H x=∑ i ∈ H,A.coordinate i x • A.unit i := by simp [projection] |
| @[simp] lemma coordinate_projection (H : Finset ι) (x : A.Space) (i : ι) : |
| A.coordinate i (A.projection H x)=if i ∈ H then A.coordinate i x else 0 := by |
| simp [projection_apply,map_sum,map_smul,smul_eq_mul,mul_ite,Finset.sum_ite_eq'] |
| lemma projection_idempotent (H : Finset ι) (x : A.Space) : |
| A.projection H (A.projection H x)=A.projection H x := by |
| rw [A.projection_apply H (A.projection H x)] |
| simp only [coordinate_projection] |
| rw [projection_apply] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [ite_eq_left hi] |
| lemma projection_commutes (H K : Finset ι) (x : A.Space) : |
| A.projection H (A.projection K x)=A.projection K (A.projection H x) := by |
| rw [A.projection_apply H (A.projection K x), |
| A.projection_apply K (A.projection H x)] |
| simp only [coordinate_projection,ite_smul,zero_smul] |
| rw [← Finset.sum_filter,← Finset.sum_filter] |
| congr 1 |
| ext i |
| simp only [Finset.mem_filter,and_comm] |
|
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| def headMap (H : Finset ι) : (H → ℝ) →ₗ[ℝ] A.Space where |
| toFun a := ∑ i : H,a i • A.unit i |
| map_add' a b := by simp [Pi.add_apply,add_smul,Finset.sum_add_distrib] |
| map_smul' t a := by simp [Pi.smul_apply,smul_smul,Finset.smul_sum] |
| lemma projection_range (H : Finset ι) : LinearMap.range (A.projection H).toLinearMap ≤ |
| LinearMap.range (A.headMap H) := by |
| rintro x ⟨y,rfl⟩ |
| refine ⟨fun i => A.coordinate i y,?_⟩ |
| change (∑ i : H,A.coordinate i y • A.unit i)=A.projection H y |
| rw [projection_apply] |
| exact Finset.sum_attach H (fun i => A.coordinate i y • A.unit i) |
|
|
| lemma projection_finite (H : Finset ι) : FiniteDimensional ℝ (LinearMap.range (A.projection H).toLinearMap) := by |
| let : FiniteDimensional ℝ (LinearMap.range (A.headMap H)) := Module.Finite.range (A.headMap H) |
| exact Submodule.finiteDimensional_of_le (A.projection_range H) |
|
|
| lemma projection_embedding (H : Finset ι) (v : ι →₀ ℝ) : |
| A.projection H (A.embedding v)=A.embedding (v.filter (· ∈ H)) := by |
| rw [projection_apply] |
| simp only [coordinate_embedding,unit] |
| simp_rw [← map_smul] |
| rw [← map_sum] |
| congr 1 |
| ext i |
| simp [Finsupp.single_apply,Finsupp.filter_apply,Finset.sum_ite_eq'] |
|
|
| lemma finite_fixed (v : ι →₀ ℝ) {H : Finset ι} (hH : v.support ⊆ H) : |
| A.projection H (A.embedding v)=A.embedding v := by |
| rw [projection_embedding] |
| congr 1 |
| ext i |
| rw [Finsupp.filter_apply] |
| split_ifs with hi |
| · rfl |
| · exact (Finsupp.notMem_support_iff.mp (fun hv => hi (hH hv))).symm |
|
|
| lemma approximation (x : A.Space) {η : ℝ} (hη : 0 < η) : |
| ∃ v : ι →₀ ℝ,‖x-A.embedding v‖ < η := by |
| obtain ⟨v,hv⟩ := UniformSpace.Completion.denseRange_coe.exists_dist_lt x hη |
| refine ⟨A.array v,?_⟩ |
| change ‖x-(v : A.Space)‖ < η |
| simpa only [dist_eq_norm] using hv |
|
|
| end Data |
| end CoordinateCompletion |
|
|
| namespace CoordinateCompletion.Data |
| variable {ι κ : Type*} (A : CoordinateCompletion.Data ι) |
|
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| def heads (H : κ → Finset ι) (hcover : ∀ S : Finset ι,∃ h,S ⊆ H h) : Midpoint.Heads κ A.Space where |
| projection h := A.projection (H h) |
| idempotent h := A.projection_idempotent (H h) |
| finite h := A.projection_finite (H h) |
| approximation x η hη := by |
| obtain ⟨v,hv⟩ := A.approximation x hη |
| obtain ⟨h,hh⟩ := hcover v.support |
| exact ⟨h,A.embedding v,A.finite_fixed v hh,hv⟩ |
| lemma heads_commutes (H : κ → Finset ι) (hcover : ∀ S : Finset ι,∃ h,S ⊆ H h) : |
| (A.heads H hcover).Commutes := by |
| intro i j x |
| exact A.projection_commutes (H i) (H j) x |
|
|
| /-- Every vector fixed by a finite head is an actual finite coordinate vector. -/ |
| lemma projection_eq_embedding (H : Finset ι) (x : A.Space) : |
| ∃ v : ι →₀ ℝ,A.projection H x=A.embedding v := by |
| refine ⟨∑ i ∈ H,(A.coordinate i x) • Finsupp.single i 1,?_⟩ |
| rw [projection_apply,map_sum] |
| simp only [map_smul,unit] |
| lemma fixed_finite {H : Finset ι} {x : A.Space} (hx : A.projection H x=x) : |
| ∃ v : ι →₀ ℝ,x=A.embedding v := by |
| obtain ⟨v,hv⟩ := A.projection_eq_embedding H x |
| exact ⟨v,hx ▸ hv⟩ |
|
|
| end CoordinateCompletion.Data |
|
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| end DiamondDistortion |
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| end |
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| end OAI |
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