| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.BlockDerivativeSum |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem linearMap_blockCoefficients {D N k : ℕ} |
| (L : (Fin (blockDimension D N k) → ℝ) →L[ℝ] ℝ) (a : Fin D → ℝ) (v : Fin (k+1) → ℝ) : |
| L (blockCoefficients (N := N) a v) = |
| (∑ i : Fin D, a i*L (Pi.single (Fin.castAdd ((k+1)*N) i) 1))+ |
| ∑ b : Fin (k+1), ∑ i : Fin N, v b*L (Pi.single (fieldIndex D N k b i) 1) := by |
| conv_lhs => rw [pi_eq_sum_univ' (blockCoefficients (N := N) a v)] |
| rw [map_sum,Fin.sum_univ_add] |
| simp only [map_smul,smul_eq_mul,blockCoefficients,Fin.addCases_left,Fin.addCases_right] |
| congr 1 |
| rw [← finProdFinEquiv.sum_comp (fun t => v (finProdFinEquiv.symm t).1*L (Pi.single (Fin.natAdd D t) 1))] |
| simp only [Equiv.symm_apply_apply,Fintype.sum_prod_type,fieldIndex_eq_natAdd] |
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| theorem field_derivative_sum {N k : ℕ} (hN : 0 < N) (E : Fin N → Fin (k+1) → ℝ) |
| (v w : Fin (k+1) → ℝ) : |
| (∑ b, ∑ i : Fin N, w b*(v b*(1-∑ l : Fin (k+1), if b ≤ l then ((k+1:ℕ):ℝ)⁻¹*E i l else 0))) = |
| (N:ℝ)*∑ b, w b*v b*(1-∑ l : Fin (k+1), if b ≤ l then ((k+1:ℕ):ℝ)⁻¹*((∑ i, E i l)/(N:ℝ)) else 0) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro b _ |
| simp only [mul_sub,mul_one,Finset.sum_sub_distrib,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul, |
| Finset.mul_sum] |
| rw [Finset.sum_comm] |
| have hn : (N:ℝ) ≠ 0 := by exact_mod_cast hN.ne' |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro l _ |
| by_cases h : b ≤ l |
| · simp only [h,ite_true] |
| simp only [← Finset.mul_sum] |
| field_simp |
| · simp only [h,ite_false,mul_zero,Finset.sum_const_zero] |
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| theorem blockPressure_field_differential {D N k : ℕ} (hN : 0 < N) |
| (I : Fin D → Finset (Fin N)) (a : Fin D → ℝ) (v w : Fin (k+1) → ℝ) : |
| fderiv ℝ (fun c : Fin (blockDimension D N k) → ℝ => |
| hierarchyPressure (blockDimension D N k) (blockMass D N k) |
| (affineLogPartition (fun _ => 0) (spinExponent (blockDimension D N k) c (blockInteraction I))) 0) |
| (blockCoefficients a v) (blockCoefficients (N := N) (fun _ => 0) w) = |
| (N:ℝ)*∑ b, w b*v b*(1-∑ l : Fin (k+1), if b ≤ l then ((k+1:ℕ):ℝ)⁻¹* |
| hierarchyMeanOverlap (blockDimension D N k) (blockMass D N k) (blockExponent I a v) |
| (fun i s => spin (s i)) (blockLevel D N k l) else 0) := by |
| rw [linearMap_blockCoefficients] |
| simp only [zero_mul,Finset.sum_const_zero,zero_add,blockPressure_coordinate_variation] |
| simp only [hierarchyMeanOverlap_eq_mean_integrals (blockDimension D N k) (blockMass D N k) (blockExponent I a v) (fun i s => spin (s i)) (fun i s => by cases s i <;> norm_num [spin])] |
| exact field_derivative_sum hN _ v w |
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| theorem blockCoefficients_hasDerivAt {D N k : ℕ} {a : ℝ → Fin D → ℝ} |
| {v : ℝ → Fin (k+1) → ℝ} {a' : Fin D → ℝ} {v' : Fin (k+1) → ℝ} {t : ℝ} |
| (ha : HasDerivAt a a' t) (hv : HasDerivAt v v' t) : |
| HasDerivAt (fun u => blockCoefficients (N := N) (a u) (v u)) (blockCoefficients a' v') t := by |
| apply hasDerivAt_pi.mpr |
| intro i |
| refine Fin.addCases (fun j => ?_) (fun j => ?_) i |
| · simpa only [blockCoefficients,Fin.addCases_left] using hasDerivAt_pi.mp ha j |
| · simpa only [blockCoefficients,Fin.addCases_right] using hasDerivAt_pi.mp hv (finProdFinEquiv.symm j).1 |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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