| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.ElementaryKernels |
|
|
| namespace OAI |
|
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| namespace ProjectionComparison |
|
|
| open Set ClusterDisplacement |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
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|
| theorem tail_projection_kernel (S : System V E) (b₀ b₁ v₀ v₁ h₀ h₁ : V → ℝ) (a : ℝ) |
| (hp₀ : IsProjection S b₀ v₀ h₀) (hp₁ : IsProjection S b₁ v₁ h₁) |
| (hb : ∀x∈S.pins,b₁ x-b₀ x≤a) : |
| ∃ A : Kernel V,Firm A ∧ ∃ r : V → ℝ,(∀x,r x∈Icc (0:ℝ) 1) ∧ |
| ∀x,max (h₁ x-h₀ x-a) 0=r x*A.apply (fun y => max (v₁ y-v₀ y-a) 0) x := by |
| classical |
| let u : V → ℝ := fun y => max (v₁ y-v₀ y-a) 0 |
| let v : V → ℝ := fun y => v₀ y+u y |
| let hne : (feasible S b₀).Nonempty := ⟨h₀,hp₀.1⟩ |
| let k := project S b₀ hne v |
| have hpk : IsProjection S b₀ v k := project_spec S b₀ hne v |
| obtain ⟨A,hA,heq⟩:=projection_update_kernel S b₀ v₀ v h₀ k hp₀ hpk |
| have he (x) : A.apply u x=k x-h₀ x := by |
| simpa only [v,add_sub_cancel_left] using congrFun heq x |
| have horder : ∀x,h₁ x≤k x+a := projection_order S b₁ b₀ v₁ v h₁ k a hp₁ hpk |
| (fun x hx => by linarith [hb x hx]) |
| (fun x hx => by dsimp [v,u];linarith [le_max_left (v₁ x-v₀ x-a) 0]) |
| have hnn (x) : 0≤A.apply u x := Finset.sum_nonneg fun y _ => mul_nonneg (A.nonneg x y) (le_max_right _ _) |
| have hle (x) : max (h₁ x-h₀ x-a) 0≤A.apply u x := by |
| apply max_le |
| · rw [he];linarith [horder x] |
| · exact hnn x |
| let r : V → ℝ := fun x => max (h₁ x-h₀ x-a) 0/A.apply u x |
| refine ⟨A,hA,r,?_,?_⟩ |
| · intro x |
| constructor |
| · exact div_nonneg (le_max_right _ _) (hnn x) |
| · by_cases hz : A.apply u x=0 |
| · simp [r,hz] |
| · exact (div_le_one (lt_of_le_of_ne (hnn x) (Ne.symm hz))).mpr (hle x) |
| · intro x |
| by_cases hz : A.apply u x=0 |
| · have hu : max (h₁ x-h₀ x-a) 0=0 := le_antisymm (by simpa only [hz] using hle x) (le_max_right _ _) |
| simp only [hu,r,zero_div,zero_mul] |
| · exact (div_mul_cancel₀ _ hz).symm |
|
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| end ProjectionComparison |
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| end OAI |
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