openAI-math-dataset / lean /OAI /Probability /LogConcave /OraclePrograms /SlotTransformAddShift.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.OraclePrograms.Expression | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open MeasureTheory | |
| open OracleCompiler OracleCompiler.Expression | |
| namespace SeedCompiler | |
| variable {k d : ℕ} | |
| lemma slotTransform_add_shift (c : ℝ) (a v : Fin k → ℝ) (g : Fin k → Point d) (Δ : Point d) : | |
| slotTransform c a (shiftSlots v Δ g)= | |
| shiftSlots (slotTransform c a v) Δ (slotTransform c a g) := by | |
| have hs : slotSum a (shiftSlots v Δ g)=slotSum a g+(slotSum a v) • Δ := by | |
| simp only [slotSum,shiftSlots,smul_add,Finset.sum_add_distrib,smul_smul,smul_eq_mul] | |
| rw [Finset.sum_smul] | |
| ext i | |
| simp only [slotTransform,shiftSlots,hs,smul_add,sub_smul,smul_smul,smul_eq_mul] | |
| abel_nf | |
| lemma slotTransform_scalar_mul (c s : ℝ) (a v : Fin k → ℝ) : | |
| slotTransform c a (fun i => s*v i)=fun i => s*slotTransform c a v i := by | |
| ext i | |
| simp only [slotTransform,slotSum,smul_eq_mul] | |
| have he : (∑i,a i*(s*v i))=s*∑i,a i*v i := by | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ring | |
| rw [he] | |
| ring | |
| lemma inverse_shift_identity (a v : Fin k → ℝ) (ha : (∑i,a i^2)<1) | |
| (s : ℝ) (g : Fin k → Point d) (Δ : Point d) : | |
| slotTransform (sqrtCoefficient a) a | |
| (shiftSlots (fun i => s*slotTransform (inverseCoefficient a) a v i) Δ g)= | |
| shiftSlots v (s • Δ) (slotTransform (sqrtCoefficient a) a g) := by | |
| rw [slotTransform_add_shift,slotTransform_scalar_mul,slotTransform_inverse a ha] | |
| funext i | |
| simp only [shiftSlots,smul_smul] | |
| rw [mul_comm s (v i)] | |
| lemma inverse_shift_energy (a v : Fin k → ℝ) (ha : (∑i,a i^2)≤1/4) (s : ℝ) : | |
| (∑i,(s*slotTransform (inverseCoefficient a) a v i)^2) ≤ | |
| 4*s^2*(∑i,v i^2) := by | |
| have h := inverse_slot_energy (E:=ℝ) a ha v | |
| simp only [Real.norm_eq_abs,sq_abs] at h | |
| simp_rw [mul_pow] | |
| rw [←Finset.mul_sum] | |
| nlinarith [mul_le_mul_of_nonneg_left h (sq_nonneg s)] | |
| lemma measurable_slotTransform (c : ℝ) (a : Fin k → ℝ) : | |
| Measurable (slotTransform (E:=Point d) c a) := by | |
| unfold slotTransform slotSum | |
| fun_prop | |
| theorem transformed_mean_equivariant {queryCount : ℕ} | |
| (M : Program (Point d × (Fin k → Point d)) d queryCount (Point d)) | |
| {r : ℝ} (hr : r≠0) (v a : Fin k → ℝ) (ha : (∑i,a i^2)<1) | |
| (hM : ∀Δ,M.Equivariant (moveSeed (fun x => x+r • Δ) v Δ) id) | |
| (c : ℝ) (Δ : Point d) : | |
| (M.seedMap (fun z => (z.1,slotTransform (sqrtCoefficient a) a z.2)) | |
| (measurable_fst.prodMk ((measurable_slotTransform _ _).comp measurable_snd))).Equivariant | |
| (moveSeed (fun x => x+c • Δ) | |
| (fun i => (c/r)*slotTransform (inverseCoefficient a) a v i) Δ) id := by | |
| apply (hM ((c/r) • Δ)).seedMap | |
| intro z | |
| apply Prod.ext | |
| · dsimp [moveSeed] | |
| rw [smul_smul] | |
| congr 2 | |
| field_simp | |
| · dsimp [moveSeed] | |
| exact inverse_shift_identity a v ha (c/r) z.2 Δ | |
| end SeedCompiler | |
| end LogConcaveSampling | |
| end | |
| end | |
| end | |
| end OAI | |