lean/OAI/Analysis/C0Absorption/Operators.lean

import OAI.Analysis.C0Absorption.Tags

namespace OAI

open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace

namespace C0Absorption
noncomputable section
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty

section GeneralC0
variable {A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
  [DiscreteTopology A] [DiscreteTopology B]

def cfunOfTendsto (f : A → ℝ) (hf : Tendsto f cofinite (nhds 0)) : C₀(A,ℝ) where
  toFun := f
  continuous_toFun := continuous_of_discreteTopology
  zero_at_infty' := by simpa only [cocompact_eq_cofinite] using hf

theorem cfun_tendsto (f : C₀(A,ℝ)) : Tendsto f cofinite (nhds 0) := by
  simpa only [cocompact_eq_cofinite] using zero_at_infty f

theorem cfun_norm_apply_le {A : Type*} [TopologicalSpace A] [DiscreteTopology A]
    (f : C₀(A,ℝ)) (a : A) : |f a|≤‖f‖ := f.toBCF.norm_coe_le_norm a

theorem cfun_norm_le {A : Type*} [TopologicalSpace A] [DiscreteTopology A]
    (f : C₀(A,ℝ)) {r : ℝ} (hr : 0≤r) (h : ∀ a,|f a|≤r) : ‖f‖≤r :=
  (f.toBCF.norm_le hr).mpr h

def cfunPull (j : A → B) (hj : Tendsto j cofinite cofinite) (f : C₀(B,ℝ)) : C₀(A,ℝ) :=
  cfunOfTendsto (fun a => f (j a)) ((cfun_tendsto f).comp hj)

@[simp] theorem cfunPull_apply (j : A → B) (hj : Tendsto j cofinite cofinite) (f : C₀(B,ℝ)) (a : A) :
    cfunPull j hj f a=f (j a) := rfl

def cfunPullL (j : A → B) (hj : Tendsto j cofinite cofinite) : C₀(B,ℝ) →L[ℝ] C₀(A,ℝ) :=
  LinearMap.mkContinuous
    { toFun := cfunPull j hj
      map_add' := by intro f g; ext a; rfl
      map_smul' := by intro c f; ext a; rfl } 1 (fun f => by
        change ‖cfunPull j hj f‖≤1*‖f‖
        rw [one_mul]
        exact cfun_norm_le _ (norm_nonneg _) (fun a => cfun_norm_apply_le f (j a)))

@[simp] theorem cfunPullL_apply (j : A → B) (hj : Tendsto j cofinite cofinite) (f : C₀(B,ℝ)) (a : A) :
    cfunPullL j hj f a=f (j a) := rfl

end GeneralC0

theorem tendsto_sumElim_cofinite {A B : Type*} {X : Type*} (f : A → X) (g : B → X)
    (l : Filter X) (hf : Tendsto f cofinite l) (hg : Tendsto g cofinite l) :
    Tendsto (Sum.elim f g) cofinite l := by
  intro s hs
  have h1 : {a | f a∉s}.Finite := hf hs
  have h2 : {b | g b∉s}.Finite := hg hs
  apply ((h1.image Sum.inl).union (h2.image Sum.inr)).subset
  intro x hx
  cases x with
  | inl a => exact Or.inl ⟨a,hx,rfl⟩
  | inr b => exact Or.inr ⟨b,hx,rfl⟩

abbrev BlockC0 := C₀(Label,ℝ)

def inputRows : C0 →L[ℝ] C0 := cfunPullL rowIndex rowIndex_injective.tendsto_cofinite

def inputBlocks : C0 →L[ℝ] BlockC0 := cfunPullL labelIndex labelIndex_injective.tendsto_cofinite

@[simp] theorem inputRows_apply (x : C0) (i : ℕ) : inputRows x i=x (rowIndex i) := rfl
@[simp] theorem inputBlocks_apply (x : C0) (γ : Label) : inputBlocks x γ=x (labelIndex γ) := rfl

def joinInputs (x : C0 × BlockC0) : C0 :=
  cfunOfTendsto (fun k => Sum.elim x.1 x.2 (coordinateEnumeration.symm k))
    ((tendsto_sumElim_cofinite x.1 x.2 (nhds 0) (cfun_tendsto x.1) (cfun_tendsto x.2)).comp
      coordinateEnumeration.symm.injective.tendsto_cofinite)

@[simp] theorem joinInputs_row (x : C0 × BlockC0) (i : ℕ) : joinInputs x (rowIndex i)=x.1 i := by
  change Sum.elim x.1 x.2 (coordinateEnumeration.symm (coordinateEnumeration (Sum.inl i)))=x.1 i
  rw [Equiv.symm_apply_apply]
  rfl

@[simp] theorem joinInputs_block (x : C0 × BlockC0) (γ : Label) : joinInputs x (labelIndex γ)=x.2 γ := by
  change Sum.elim x.1 x.2 (coordinateEnumeration.symm (coordinateEnumeration (Sum.inr γ)))=x.2 γ
  rw [Equiv.symm_apply_apply]
  rfl

theorem joinInputs_split (x : C0) : joinInputs (inputRows x,inputBlocks x)=x := by
  ext k
  obtain ⟨a,rfl⟩ := coordinateEnumeration.surjective k
  cases a with
  | inl i => exact joinInputs_row _ i
  | inr γ => exact joinInputs_block _ γ

theorem joinInputs_norm (x : C0 × BlockC0) : ‖joinInputs x‖=‖x‖ := by
  apply le_antisymm
  · apply cfun_norm_le _ (norm_nonneg x)
    intro k
    obtain ⟨a,rfl⟩ := coordinateEnumeration.surjective k
    cases a with
    | inl i => exact (show |joinInputs x (rowIndex i)|≤‖x.1‖ by rw [joinInputs_row]; exact cfun_norm_apply_le _ _).trans (le_max_left _ _)
    | inr γ => exact (show |joinInputs x (labelIndex γ)|≤‖x.2‖ by rw [joinInputs_block]; exact cfun_norm_apply_le _ _).trans (le_max_right _ _)
  · rw [Prod.norm_def]
    apply max_le
    · apply cfun_norm_le _ (norm_nonneg _)
      intro i
      rw [← joinInputs_row x i]
      exact cfun_norm_apply_le _ _
    · apply cfun_norm_le _ (norm_nonneg _)
      intro γ
      rw [← joinInputs_block x γ]
      exact cfun_norm_apply_le _ _

def inputCoordinates : (C0 × BlockC0) ≃ₗᵢ[ℝ] C0 where
  toFun := joinInputs
  invFun := fun x => (inputRows x,inputBlocks x)
  left_inv := by
    intro x
    apply Prod.ext <;> ext k
    · exact joinInputs_row x k
    · exact joinInputs_block x k
  right_inv := joinInputs_split
  map_add' := by
    intro x y
    ext k
    obtain ⟨a,rfl⟩ := coordinateEnumeration.surjective k
    cases a with
    | inl i => simp only [show coordinateEnumeration (Sum.inl i)=rowIndex i from rfl,joinInputs_row,Prod.fst_add,ZeroAtInftyContinuousMap.add_apply]
    | inr γ => simp only [show coordinateEnumeration (Sum.inr γ)=labelIndex γ from rfl,joinInputs_block,Prod.snd_add,ZeroAtInftyContinuousMap.add_apply]
  map_smul' := by
    intro c x
    ext k
    obtain ⟨a,rfl⟩ := coordinateEnumeration.surjective k
    cases a with
    | inl i => simp only [show coordinateEnumeration (Sum.inl i)=rowIndex i from rfl,joinInputs_row,ZeroAtInftyContinuousMap.smul_apply]; rfl
    | inr γ => simp only [show coordinateEnumeration (Sum.inr γ)=labelIndex γ from rfl,joinInputs_block,ZeroAtInftyContinuousMap.smul_apply]; rfl
  norm_map' := joinInputs_norm

theorem labelLevel_tendsto : Tendsto Label.level cofinite cofinite :=
  Filter.Tendsto.cofinite_of_finite_preimage_singleton (fun lev => labels_at_level_finite lev)

theorem labelRadius_tendsto (x : C0) : Tendsto (fun γ : Label => localRadius γ.level x) cofinite (nhds 0) :=
  (localRadius_tendsto x).comp labelLevel_tendsto

theorem frozenCorrection_tendsto (W : FrozenWeights) (x : C0) :
    Tendsto (fun γ => frozenCorrection W γ x) cofinite (nhds 0) := by
  apply squeeze_zero_norm (fun γ => frozenCorrection_local W γ x) _
  simpa only [mul_zero] using (labelRadius_tendsto x).const_mul 4

def frozenCorrectionC0 (W : FrozenWeights) (x : C0) : BlockC0 :=
  cfunOfTendsto (fun γ => frozenCorrection W γ x) (frozenCorrection_tendsto W x)

def frozenCorrectionL (W : FrozenWeights) : C0 →L[ℝ] BlockC0 :=
  LinearMap.mkContinuous
    { toFun := frozenCorrectionC0 W
      map_add' := by intro x y; ext γ; exact map_add (frozenCorrection W γ) x y
      map_smul' := by intro c x; ext γ; exact map_smul (frozenCorrection W γ) c x }
    4 (fun x => by
      change ‖frozenCorrectionC0 W x‖≤4*‖x‖
      exact cfun_norm_le _ (mul_nonneg (by norm_num) (norm_nonneg _)) (fun γ => frozenCorrection_bound W γ x))

@[simp] theorem frozenCorrectionL_apply (W : FrozenWeights) (x : C0) (γ : Label) :
    frozenCorrectionL W x γ=frozenCorrection W γ x := rfl

def correctionC0 (s : C0Ball) : BlockC0 := frozenCorrectionL (stateWeights s) s.val

@[simp] theorem correctionC0_apply (s : C0Ball) (γ : Label) : correctionC0 s γ=correction γ s := rfl

end
end C0Absorption

end OAI