| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace ProjectionComparison | |
| open Set | |
| structure System (V E : Type*) where | |
| left : E → V | |
| right : E → V | |
| cost : E → ℝ | |
| pins : Set V | |
| variable {V E : Type*} | |
| def feasible (S : System V E) (b : V → ℝ) : Set (V → ℝ) := | |
| {h | (∀ e,h (S.left e)-h (S.right e)≤S.cost e) ∧ ∀ x∈S.pins,h x=b x} | |
| def objective [Fintype V] (v h : V → ℝ) : ℝ := ∑ x,(h x-v x)^2 | |
| def IsProjection [Fintype V] (S : System V E) (b v h : V → ℝ) : Prop := | |
| h∈feasible S b ∧ ∀ k∈feasible S b,objective v h≤objective v k | |
| end ProjectionComparison | |
| namespace ProjectionComparison | |
| open Set | |
| variable {V E : Type*} | |
| lemma min_feasible (S : System V E) (b c h k : V → ℝ) (δ : ℝ) | |
| (hh : h∈feasible S b) (hk : k∈feasible S c) | |
| (hb : ∀ x∈S.pins,b x≤c x+δ) : | |
| (fun x => min (h x) (k x+δ))∈feasible S b := by | |
| constructor | |
| · intro e | |
| apply sub_le_iff_le_add.mpr | |
| have h₁ : h (S.left e)≤h (S.right e)+S.cost e := by linarith [hh.1 e] | |
| have h₂ : k (S.left e)+δ≤(k (S.right e)+δ)+S.cost e := by linarith [hk.1 e] | |
| simpa only [min_add_add_right, add_comm (S.cost e)] using min_le_min h₁ h₂ | |
| · intro x hx | |
| dsimp only | |
| rw [hh.2 x hx,hk.2 x hx,min_eq_left (hb x hx)] | |
| lemma max_feasible (S : System V E) (b c h k : V → ℝ) (δ : ℝ) | |
| (hh : h∈feasible S b) (hk : k∈feasible S c) | |
| (hb : ∀ x∈S.pins,c x+δ≤b x) : | |
| (fun x => max (h x) (k x+δ))∈feasible S b := by | |
| constructor | |
| · intro e | |
| apply sub_le_iff_le_add.mpr | |
| have h₁ : h (S.left e)≤h (S.right e)+S.cost e := by linarith [hh.1 e] | |
| have h₂ : k (S.left e)+δ≤(k (S.right e)+δ)+S.cost e := by linarith [hk.1 e] | |
| simpa only [max_add_add_right, add_comm (S.cost e)] using max_le_max h₁ h₂ | |
| · intro x hx | |
| dsimp only | |
| rw [hh.2 x hx,hk.2 x hx,max_eq_left (hb x hx)] | |
| lemma interpolate_feasible (S : System V E) (b h k : V → ℝ) | |
| (hh : h∈feasible S b) (hk : k∈feasible S b) (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : | |
| (fun x => (1-t)*h x+t*k x)∈feasible S b := by | |
| refine ⟨fun e => ?_,fun x hx => ?_⟩ | |
| · have h₁ := mul_le_mul_of_nonneg_left (hh.1 e) (by linarith [ht.2] : 0≤1-t) | |
| have h₂ := mul_le_mul_of_nonneg_left (hk.1 e) ht.1 | |
| nlinarith [h₁,h₂] | |
| · dsimp only;rw [hh.2 x hx,hk.2 x hx];ring | |
| variable [Fintype V] | |
| lemma objective_interpolate (v h k : V → ℝ) (t : ℝ) : | |
| objective v (fun x => (1-t)*h x+t*k x)=objective v h+ | |
| 2*t*(∑ x,(h x-v x)*(k x-h x))+t^2*(∑ x,(k x-h x)^2) := by | |
| unfold objective | |
| calc | |
| _ = ∑ x,((h x-v x)^2+2*t*((h x-v x)*(k x-h x))+t^2*(k x-h x)^2) := by | |
| apply Finset.sum_congr rfl;intro x hx;ring | |
| _ = _ := by simp only [Finset.sum_add_distrib,← Finset.mul_sum] | |
| lemma variational (S : System V E) (b v h : V → ℝ) (hp : IsProjection S b v h) | |
| (k : V → ℝ) (hk : k∈feasible S b) : 0≤∑ x,(h x-v x)*(k x-h x) := by | |
| let A := ∑ x,(h x-v x)*(k x-h x) | |
| let T := ∑ x,(k x-h x)^2 | |
| have hT : 0≤T := Finset.sum_nonneg fun x _ => sq_nonneg _ | |
| by_contra hA | |
| have hA' : A<0 := lt_of_not_ge hA | |
| let t := min 1 ((-A)/(T+1)) | |
| have ht : 0<t := lt_min (by norm_num) (div_pos (by linarith) (by linarith)) | |
| have ht1 : t≤1 := min_le_left _ _ | |
| have htA : t*(T+1)≤-A := (le_div_iff₀ (by linarith : 0<T+1)).mp (min_le_right _ _) | |
| have hineq := hp.2 _ (interpolate_feasible S b h k hp.1 hk t ⟨ht.le,ht1⟩) | |
| rw [objective_interpolate] at hineq | |
| change objective v h≤objective v h+2*t*A+t^2*T at hineq | |
| have hn : 2*A+t*T<0 := by linarith | |
| have hneg := mul_neg_of_pos_of_neg ht hn | |
| nlinarith [hneg] | |
| lemma projection_order (S : System V E) (b₁ b₂ v₁ v₂ h₁ h₂ : V → ℝ) (δ : ℝ) | |
| (hp₁ : IsProjection S b₁ v₁ h₁) (hp₂ : IsProjection S b₂ v₂ h₂) | |
| (hb : ∀ x∈S.pins,b₁ x≤b₂ x+δ) (hv : ∀ x∉S.pins,v₁ x≤v₂ x+δ) : | |
| ∀ x,h₁ x≤h₂ x+δ := by | |
| let H : V → ℝ := fun x => max (h₁ x-h₂ x-δ) 0 | |
| have hH (x) : 0≤H x := le_max_right _ _ | |
| let k₁ : V → ℝ := fun x => min (h₁ x) (h₂ x+δ) | |
| let k₂ : V → ℝ := fun x => max (h₂ x) (h₁ x+(-δ)) | |
| have hk₁ := min_feasible S b₁ b₂ h₁ h₂ δ hp₁.1 hp₂.1 hb | |
| have hk₂ := max_feasible S b₂ b₁ h₂ h₁ (-δ) hp₂.1 hp₁.1 (fun x hx => by linarith [hb x hx]) | |
| have ht₁ := variational S b₁ v₁ h₁ hp₁ k₁ hk₁ | |
| have ht₂ := variational S b₂ v₂ h₂ hp₂ k₂ hk₂ | |
| have hd₁ (x) : k₁ x-h₁ x=-(H x) := by | |
| dsimp [k₁,H];by_cases hx : h₁ x≤h₂ x+δ | |
| · rw [min_eq_left hx,max_eq_right (by linarith)];ring | |
| · rw [min_eq_right (by linarith),max_eq_left (by linarith)];ring | |
| have hd₂ (x) : k₂ x-h₂ x=H x := by | |
| dsimp [k₂,H];by_cases hx : h₁ x≤h₂ x+δ | |
| · rw [max_eq_left (by linarith),max_eq_right (by linarith)];ring | |
| · rw [max_eq_right (by linarith),max_eq_left (by linarith)];ring | |
| simp only [hd₁,mul_neg,Finset.sum_neg_distrib,neg_nonneg] at ht₁ | |
| simp only [hd₂] at ht₂ | |
| have hpoint (x : V) : (H x)^2≤(h₁ x-v₁ x)*H x-(h₂ x-v₂ x)*H x := by | |
| by_cases hx : x∈S.pins | |
| · have hz : H x=0 := by dsimp [H];rw [hp₁.1.2 x hx,hp₂.1.2 x hx,max_eq_right (by linarith [hb x hx])] | |
| rw [hz];ring_nf;exact le_rfl | |
| · have hvx := hv x hx | |
| dsimp [H] | |
| by_cases hm : h₁ x-h₂ x-δ≤0 | |
| · rw [max_eq_right hm];simp | |
| · rw [max_eq_left (by linarith)] | |
| nlinarith [mul_nonneg (by linarith : 0≤h₁ x-h₂ x-δ) (by linarith : 0≤v₂ x+δ-v₁ x)] | |
| have hsum : (∑ x,(H x)^2)≤0 := by | |
| have h := Finset.sum_le_sum (fun x (_ : x∈Finset.univ) => hpoint x) | |
| rw [Finset.sum_sub_distrib] at h | |
| linarith | |
| intro x | |
| have hx := Finset.single_le_sum (fun y (_ : y∈Finset.univ) => sq_nonneg (H y)) (Finset.mem_univ x) | |
| have hz : H x=0 := by nlinarith [hx,hsum] | |
| have hm := le_max_left (h₁ x-h₂ x-δ) 0 | |
| change h₁ x-h₂ x-δ≤H x at hm | |
| linarith | |
| theorem projection_parameter_stability (S : System V E) (b₁ b₂ v₁ v₂ h₁ h₂ : V → ℝ) (δ : ℝ) (_hδ : 0≤δ) | |
| (hp₁ : IsProjection S b₁ v₁ h₁) (hp₂ : IsProjection S b₂ v₂ h₂) | |
| (hb : ∀ x∈S.pins,|b₁ x-b₂ x|≤δ) (hv : ∀ x∉S.pins,|v₁ x-v₂ x|≤δ) : | |
| ∀ x,|h₁ x-h₂ x|≤δ := by | |
| have hup := projection_order S b₁ b₂ v₁ v₂ h₁ h₂ δ hp₁ hp₂ | |
| (fun x hx => by have := (abs_le.mp (hb x hx)).2;linarith) | |
| (fun x hx => by have := (abs_le.mp (hv x hx)).2;linarith) | |
| have hlo := projection_order S b₂ b₁ v₂ v₁ h₂ h₁ δ hp₂ hp₁ | |
| (fun x hx => by have := (abs_le.mp (hb x hx)).1;linarith) | |
| (fun x hx => by have := (abs_le.mp (hv x hx)).1;linarith) | |
| intro x | |
| exact abs_le.mpr ⟨by linarith [hlo x],by linarith [hup x]⟩ | |
| end ProjectionComparison | |
| end | |
| end OAI | |