| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionPattern |
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| namespace OAI |
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| namespace ProjectionComparison |
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| open Set MeasureTheory |
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| variable {V E : Type*} [Fintype V] [Fintype E] |
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| theorem projection_mixture (S : System V E) (b v₀ v₁ h₀ h₁ : V → ℝ) |
| (hp₀ : IsProjection S b v₀ h₀) (hp₁ : IsProjection S b v₁ h₁) : |
| ∃ w : Finset E → ℝ,(∀ F,0≤w F) ∧ (∑ F,w F)=1 ∧ |
| ∀ x,h₁ x-h₀ x=∑ F,w F*faceAverage S F (fun y => v₁ y-v₀ y) x := by |
| classical |
| let hne : (feasible S b).Nonempty := ⟨h₀,hp₀.1⟩ |
| let H : ℝ → V → ℝ := fun t => project S b hne (fun y => v₀ y+t*(v₁ y-v₀ y)) |
| have hlip (x : V) : LipschitzWith ‖v₁-v₀‖₊ (fun t => H t x) := project_path_lipschitz S b v₀ v₁ hne x |
| have hc (x : V) : Continuous (fun t => H t x) := (hlip x).continuous |
| have hH₀ : H 0=h₀ := by |
| apply projection_unique S b v₀ _ h₀ _ hp₀ |
| simpa only [H,zero_mul,add_zero] using project_spec S b hne v₀ |
| have hH₁ : H 1=h₁ := by |
| apply projection_unique S b v₁ _ h₁ _ hp₁ |
| simpa only [H,one_mul,add_sub_cancel] using project_spec S b hne v₁ |
| have hder : ∀ᵐ t : ℝ,∀ x,deriv (fun s => H s x) t=faceAverage S (activePattern S (H t)) (fun y => v₁ y-v₀ y) x := by |
| filter_upwards [project_path_ae_derivative S b v₀ v₁ hne] with t ht |
| have hd := projection_derivative S (fun _ => b) (fun s y => v₀ y+s*(v₁ y-v₀ y)) H |
| (fun _ => 0) (fun y => v₁ y-v₀ y) (fun y => deriv (fun s => H s y) t) t |
| (Filter.Eventually.of_forall fun s => project_spec S b hne _) |
| (fun y => hasDerivAt_const t (b y)) |
| (fun y => by simpa using (((hasDerivAt_id t).mul_const (v₁ y-v₀ y)).const_add (v₀ y))) |
| (fun y => (ht y).hasDerivAt) |
| intro x |
| unfold faceAverage |
| rw [faceGraph_pattern,faceComponent_pattern] |
| split_ifs with hf |
| · exact hd.2 x hf |
| · push Not at hf |
| obtain ⟨p,hp,hxp⟩ := hf |
| exact hd.1 x p hp hxp |
| let w : Finset E → ℝ := fun F => ∫ t in (0:ℝ)..1,patternWeight S H F t |
| refine ⟨w,?_,?_,?_⟩ |
| · intro F |
| apply intervalIntegral.integral_nonneg (by norm_num) |
| intro t ht |
| unfold patternWeight |
| split_ifs <;> norm_num |
| · dsimp only [w] |
| rw [← intervalIntegral.integral_finsetSum (fun F _ => intervalIntegrable_patternWeight S H hc F)] |
| simp only [sum_patternWeight] |
| norm_num |
| · intro x |
| have ha : AbsolutelyContinuousOnInterval (fun t => H t x) 0 1 := (hlip x).lipschitzOnWith.absolutelyContinuousOnInterval |
| calc |
| h₁ x-h₀ x=H 1 x-H 0 x := by rw [hH₀,hH₁] |
| _ = ∫ t in (0:ℝ)..1,deriv (fun s => H s x) t := ha.integral_deriv_eq_sub.symm |
| _ = ∫ t in (0:ℝ)..1,∑ F,patternWeight S H F t*faceAverage S F (fun y => v₁ y-v₀ y) x := by |
| apply intervalIntegral.integral_congr_ae |
| filter_upwards [hder] with t ht |
| intro _ |
| rw [sum_patternWeight_mul] |
| exact ht x |
| _ = ∑ F,w F*faceAverage S F (fun y => v₁ y-v₀ y) x := by |
| rw [intervalIntegral.integral_finsetSum (fun F _ => (intervalIntegrable_patternWeight S H hc F).mul_const _)] |
| simp only [intervalIntegral.integral_mul_const,w] |
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| end ProjectionComparison |
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| end OAI |
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