| import Mathlib |
| import OAI.Probability.SKGap.Localization.Taylor |
| import OAI.Probability.SKGap.Localization.ScalarVariance |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set |
| namespace SKGapCutoff.Recipe |
| variable {n : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| def flipHalfDiff (i : Fin n) (P : Spin n→E) (x : Spin n) : E := |
| (spin x i/2) • (P x-P (flip x i)) |
|
|
| lemma flipHalfDiff_norm (i : Fin n) (P : Spin n→E) (x : Spin n) : |
| ‖flipHalfDiff i P x‖=‖P (flip x i)-P x‖/2 := by |
| simp only [flipHalfDiff,norm_smul,Real.norm_eq_abs,abs_div,abs_spin_eq_one,abs_of_pos (by norm_num : (0:ℝ)<2)] |
| rw [norm_sub_rev] |
| ring |
|
|
| lemma flipHalfDiff_real (i : Fin n) (P : Spin n→ℝ) (x : Spin n) : |
| flipHalfDiff i P x=halfDiff i P x := by |
| rw [halfDiff_as_flip] |
| dsimp only [flipHalfDiff,smul_eq_mul] |
| ring |
|
|
| lemma coefficient_linearization (i : Fin n) (P : Spin n→E) (x : Spin n) |
| (F : E→ℝ) (T : E→L[ℝ]ℝ) : |
| halfDiff i (fun y=>F (P y)) x-T (flipHalfDiff i P x)= |
| (-spin x i/2)*(F (P (flip x i))-F (P x)-T (P (flip x i)-P x)) := by |
| rw [halfDiff_as_flip] |
| simp only [flipHalfDiff,map_smul,map_sub,smul_eq_mul] |
| ring |
|
|
| lemma coefficient_taylor (i : Fin n) (P : Spin n→E) (x : Spin n) |
| (F : E→ℝ) (F' : E→E→L[ℝ]ℝ) {L : ℝ} (hL : 0≤L) |
| (hF : ∀t∈Icc (0:ℝ) 1,HasFDerivAt F |
| (F' (P x+t • (P (flip x i)-P x))) (P x+t • (P (flip x i)-P x))) |
| (hLip : ∀t∈Icc (0:ℝ) 1, |
| ‖F' (P x+t • (P (flip x i)-P x))-F' (P x)‖≤L*‖t • (P (flip x i)-P x)‖) : |
| |halfDiff i (fun y=>F (P y)) x-F' (P x) (flipHalfDiff i P x)|≤ |
| 2*L*‖flipHalfDiff i P x‖^2 := by |
| have H:=SKGap.SourceTaylor.segment_taylor F F' (P x) (P (flip x i)) hL hF hLip |
| rw [coefficient_linearization,abs_mul,abs_div,abs_neg,abs_spin_eq_one, |
| abs_of_pos (by norm_num : (0:ℝ)<2),flipHalfDiff_norm] |
| calc |
| _ ≤ (1/2:ℝ)*(L*‖P (flip x i)-P x‖^2) := mul_le_mul_of_nonneg_left H (by norm_num) |
| _ = _ := by ring |
|
|
| lemma coefficient_lipschitz (i : Fin n) (P : Spin n→E) (x : Spin n) |
| (F : E→ℝ) (F' : E→E→L[ℝ]ℝ) {C : ℝ} (hC : 0≤C) |
| (hF : ∀t∈Icc (0:ℝ) 1,HasFDerivAt F |
| (F' (P x+t • (P (flip x i)-P x))) (P x+t • (P (flip x i)-P x))) |
| (hBound : ∀t∈Icc (0:ℝ) 1,‖F' (P x+t • (P (flip x i)-P x))‖≤C) : |
| |halfDiff i (fun y=>F (P y)) x|≤C*‖flipHalfDiff i P x‖ := by |
| have H:=SKGap.SourceTaylor.segment_lipschitz F F' (P x) (P (flip x i)) hC hF hBound |
| rw [halfDiff_as_flip,abs_div,abs_mul,abs_spin_eq_one,one_mul, |
| abs_of_pos (by norm_num : (0:ℝ)<2),flipHalfDiff_norm,abs_sub_comm] |
| simpa only [mul_div_assoc] using div_le_div_of_nonneg_right H (show (0:ℝ)≤2 by norm_num) |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| end OAI |
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|
