File size: 3,819 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Probability.ClassicalON.BlockKernel
namespace OAI
universe uE uV
noncomputable section
open MeasureTheory
open scoped InnerProductSpace BigOperators Classical
namespace ClassicalON
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E]
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
end ClassicalON
end
end OAI
|