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dd14c26 | 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 | import Mathlib
import OAI.Probability.LatticeSurface.Dynamics.ProjectionPattern
namespace OAI
namespace ProjectionComparison
open Set MeasureTheory
variable {V E : Type*} [Fintype V] [Fintype E]
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]
end ProjectionComparison
end OAI
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