| import OAI.Probability.ClassicalON.BlockKernel |
| |
| namespace OAI |
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| universe uE uV |
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| noncomputable section |
| open MeasureTheory |
| open scoped InnerProductSpace BigOperators Classical |
| namespace ClassicalON |
| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] |
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| theorem coordinateSpinMean_block_bounds (k : ℕ) (left right : E → V) (b : E → ℝ) |
| (hb : ∀ e,0 ≤ b e) (x y : V) (C : ℝ) |
| (hthree : ∀ c : E → ℝ,(∀ e,0 ≤ c e ∧ c e ≤ b e) → |
| 0 ≤ freeSpinMean 3 left right c (firstSpinProduct x y) ∧ |
| freeSpinMean 3 left right c (firstSpinProduct x y) ≤ C) : |
| 0 ≤ freeSpinMean (3+k) left right b (coordinateSpinProduct (3+k) ⟨0,by omega⟩ x y) ∧ |
| freeSpinMean (3+k) left right b (coordinateSpinProduct (3+k) ⟨0,by omega⟩ x y) ≤ C := by |
| let : NeZero (3+k) := ⟨by omega⟩ |
| let W : (V → Spin (3+k)) → ℝ := fun s => Real.exp (freeSpinEnergy (3+k) left right b s) |
| let F := coordinateSpinProduct (3+k) ⟨0,by omega⟩ x y |
| have hW : Continuous W := (continuous_freeSpinEnergy _ _ _ _).rexp |
| have hF : Continuous F := continuous_coordinateSpinProduct _ _ _ _ |
| have hWF : Continuous (fun s => W s*F s) := hW.mul hF |
| have hpoint (s : V → Spin (3+k)) : |
| 0 ≤ (∫ t,W (blockFiberConfiguration k s t)*F (blockFiberConfiguration k s t) ∂freeSpinReference 3) ∧ |
| (∫ t,W (blockFiberConfiguration k s t)*F (blockFiberConfiguration k s t) ∂freeSpinReference 3) ≤ |
| C*(∫ t,W (blockFiberConfiguration k s t) ∂freeSpinReference 3) := by |
| dsimp only [W,F] |
| rw [block_fiber_numerator,block_fiber_partition] |
| have h3 := hthree (blockCoupling k left right b s) (blockCoupling_bounds k left right b hb s) |
| have hr0 : 0 ≤ blockRadius k (s x)*blockRadius k (s y) := |
| mul_nonneg (blockRadius_nonneg _ _) (blockRadius_nonneg _ _) |
| have hr1 : blockRadius k (s x)*blockRadius k (s y) ≤ 1 := |
| (mul_le_of_le_one_left (blockRadius_nonneg _ _) (blockRadius_le_one _ _)).trans |
| (blockRadius_le_one _ _) |
| have hD : 0 ≤ Real.exp (blockTailEnergy k left right b s)* |
| freeSpinPartition 3 left right (blockCoupling k left right b s) := |
| mul_nonneg (Real.exp_pos _).le (freeSpinPartition_pos _ _ _ _).le |
| constructor |
| · exact mul_nonneg hD (mul_nonneg hr0 h3.1) |
| · calc |
| _ ≤ (Real.exp (blockTailEnergy k left right b s)* |
| freeSpinPartition 3 left right (blockCoupling k left right b s))*C := |
| mul_le_mul_of_nonneg_left |
| ((mul_le_of_le_one_left h3.1 hr1).trans h3.2) hD |
| _ = _ := mul_comm _ _ |
| have hNi : Integrable (fun s : V → Spin (3+k) => |
| ∫ t,W (blockFiberConfiguration k s t)*F (blockFiberConfiguration k s t) ∂freeSpinReference 3) |
| (freeSpinReference (3+k)) := |
| compact_integrable (compact_parametric_integral _ |
| (hWF.comp (continuous_blockFiberConfiguration k))) |
| have hDi : Integrable (fun s : V → Spin (3+k) => |
| ∫ t,W (blockFiberConfiguration k s t) ∂freeSpinReference 3) (freeSpinReference (3+k)) := |
| compact_integrable (compact_parametric_integral _ |
| (hW.comp (continuous_blockFiberConfiguration k))) |
| have hN : 0 ≤ ∫ s,W s*F s ∂freeSpinReference (3+k) := by |
| rw [integral_freeSpinReference_block k _ hWF] |
| exact integral_nonneg (fun s => (hpoint s).1) |
| have hND : (∫ s,W s*F s ∂freeSpinReference (3+k)) ≤ C*(∫ s,W s ∂freeSpinReference (3+k)) := by |
| rw [integral_freeSpinReference_block k _ hWF,integral_freeSpinReference_block k _ hW,← integral_const_mul] |
| exact integral_mono hNi (hDi.const_mul C) (fun s => (hpoint s).2) |
| have hD : 0 < ∫ s,W s ∂freeSpinReference (3+k) := |
| compact_integral_pos hW (fun _ => Real.exp_pos _) |
| constructor |
| · exact div_nonneg hN hD.le |
| · exact (div_le_iff₀ hD).mpr hND |
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| end ClassicalON |
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| end |
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| end OAI |
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