| import Mathlib |
| import OAI.Probability.SKRatio.Calculus.PlusSpin |
| import OAI.Probability.SKRatio.Variational.ScalarKernel |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Set Real |
| namespace SKRatio.Planted |
| open Scalar Calculus |
|
|
| abbrev FieldWeight := Set.Icc (0:ℝ) 2 |
| |
| instance : CompactSpace FieldWeight := inferInstanceAs (CompactSpace (Set.Icc (0:ℝ) 2)) |
| |
| def compactWeight (h : ℝ) : FieldWeight := |
| ⟨w h, (w_pos h).le, by unfold w; linarith [(m_bounds h).1]⟩ |
| |
| lemma compactWeight_lipschitz : LipschitzWith 1 compactWeight := by |
| apply LipschitzWith.of_dist_le_mul |
| intro h t |
| change |w h-w t| ≤ (1:ℝ)*|h-t| |
| simpa only [w,m,sub_sub_sub_cancel_left,abs_sub_comm,one_mul] using tanh_lipschitz_abs h t |
| |
| @[simp] lemma compactWeight_coe (h : ℝ) : (compactWeight h : ℝ) = w h := rfl |
| |
| def compactV (a : FieldWeight) : ℝ := (a:ℝ)*(2-a) |
| |
| def compactF (p : FieldWeight × FieldWeight) : ℝ := |
| (p.2:ℝ)^2*(2-p.2)^2/((p.1:ℝ)+p.2) |
| def compactK (p : FieldWeight × FieldWeight) : ℝ := |
| compactV p.1*compactV p.2/((p.1:ℝ)+p.2) |
| |
| lemma compactV_nonneg (a : FieldWeight) : 0≤compactV a := |
| mul_nonneg a.2.1 (sub_nonneg.mpr a.2.2) |
| lemma compactV_le_one (a : FieldWeight) : compactV a≤1 := by |
| dsimp [compactV] |
| nlinarith [sq_nonneg ((a:ℝ)-1)] |
| lemma compactV_le (a : FieldWeight) : compactV a≤2*a := by |
| dsimp [compactV] |
| nlinarith [sq_nonneg (a:ℝ)] |
| lemma compactF_nonneg (p : FieldWeight × FieldWeight) : 0≤compactF p := by |
| unfold compactF |
| exact div_nonneg (mul_nonneg (sq_nonneg _) (sq_nonneg _)) (add_nonneg p.1.2.1 p.2.2.1) |
| lemma compactK_nonneg (p : FieldWeight × FieldWeight) : 0≤compactK p := by |
| unfold compactK |
| exact div_nonneg (mul_nonneg (compactV_nonneg _) (compactV_nonneg _)) (add_nonneg p.1.2.1 p.2.2.1) |
| |
| lemma compactF_bound (p : FieldWeight × FieldWeight) : compactF p ≤ 4*(p.2:ℝ) := by |
| by_cases hz : (p.1:ℝ)+(p.2:ℝ)=0 |
| · simp only [compactF,hz,div_zero] |
| exact mul_nonneg (by norm_num) p.2.2.1 |
| have hd : 0<(p.1:ℝ)+(p.2:ℝ) := lt_of_le_of_ne (add_nonneg p.1.2.1 p.2.2.1) (Ne.symm hz) |
| apply (div_le_iff₀ hd).mpr |
| have hs : (2-(p.2:ℝ))^2≤4 := by nlinarith [p.2.2.1,p.2.2.2] |
| have hm := mul_le_mul_of_nonneg_left hs (sq_nonneg (p.2:ℝ)) |
| nlinarith only [hm,mul_nonneg p.1.2.1 p.2.2.1] |
| |
| lemma compactK_bound (p : FieldWeight × FieldWeight) : compactK p ≤ 4*(p.2:ℝ) := by |
| by_cases hz : (p.1:ℝ)+(p.2:ℝ)=0 |
| · simp only [compactK,hz,div_zero] |
| exact mul_nonneg (by norm_num) p.2.2.1 |
| have hd : 0<(p.1:ℝ)+(p.2:ℝ) := lt_of_le_of_ne (add_nonneg p.1.2.1 p.2.2.1) (Ne.symm hz) |
| apply (div_le_iff₀ hd).mpr |
| have hm := mul_le_mul (compactV_le p.1) (compactV_le p.2) |
| (compactV_nonneg p.2) (mul_nonneg (by norm_num : (0:ℝ)≤2) p.1.2.1) |
| nlinarith only [hm,sq_nonneg (p.2:ℝ)] |
| |
| lemma continuous_compactV : Continuous compactV := by unfold compactV; fun_prop |
| |
| lemma continuous_compact_div {f : FieldWeight × FieldWeight → ℝ} |
| (hnonneg : ∀ p, 0≤f p) (hbound : ∀ p, f p≤4*(p.2:ℝ)) |
| (hoff : ∀ p, (p.1:ℝ)+(p.2:ℝ)≠0 → ContinuousAt f p) : Continuous f := by |
| apply continuous_iff_continuousAt.mpr |
| intro p |
| by_cases hz : (p.1:ℝ)+(p.2:ℝ)=0 |
| · have h2 : (p.2:ℝ)=0 := by linarith [p.1.2.1,p.2.2.1] |
| have hp : f p=0 := le_antisymm (by simpa [h2] using hbound p) (hnonneg p) |
| rw [ContinuousAt,hp] |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds |
| (h := fun q : FieldWeight × FieldWeight => 4*(q.2:ℝ)) |
| (by simpa only [ContinuousAt,h2,mul_zero] using |
| (show Continuous (fun q : FieldWeight × FieldWeight => 4*(q.2:ℝ)) by fun_prop).continuousAt (x := p)) |
| hnonneg hbound |
| · exact hoff p hz |
| |
| lemma continuous_compactF : Continuous compactF := by |
| apply continuous_compact_div compactF_nonneg compactF_bound |
| intro p hp |
| exact ContinuousAt.div (by fun_prop) (by fun_prop) hp |
| |
| lemma continuous_compactK : Continuous compactK := by |
| apply continuous_compact_div compactK_nonneg compactK_bound |
| intro p hp |
| exact ContinuousAt.div |
| ((continuous_compactV.comp continuous_fst).mul (continuous_compactV.comp continuous_snd)).continuousAt |
| (by fun_prop) hp |
| |
| lemma compactV_weight (h : ℝ) : compactV (compactWeight h) = v h := by |
| unfold compactV compactWeight w v |
| dsimp |
| ring |
| |
| lemma compactF_weight (h t : ℝ) : compactF (compactWeight h,compactWeight t) = |
| v t^2/(w h+w t) := by |
| unfold compactF |
| simp only [compactWeight_coe,←mul_pow] |
| rw [show w t*(2-w t)=v t from compactV_weight t] |
| |
| lemma compactK_weight (h t : ℝ) : compactK (compactWeight h,compactWeight t) = |
| kernel h t := by |
| unfold compactK kernel |
| rw [compactV_weight,compactV_weight] |
| rfl |
| |
| end SKRatio.Planted |
| |
| end |
| end |
| |
| end OAI |
| |
