| import OAI.Probability.InvariantIsing.Magnetic.MagneticPopulationCoupling |
|
|
| namespace OAI |
|
|
| /-! The field contribution changes by at most the transportation cost |
| when an interior profile is averaged across field populations. -/ |
| noncomputable section |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma transportedMagnetization_field_term {A B : Type*} [Fintype A] [Fintype B] |
| (w : A → B → ℝ) (δ c : B → ℝ) (mag : A → ℝ) (hδ : ∀ b, 0 < δ b) : |
| (∑ b, δ b*c b*transportedMagnetization w δ mag b)= |
| ∑ a, ∑ b, w a b*c b*mag a := by |
| calc |
| _ = ∑ b, c b*(∑ a, w a b*mag a) := by |
| apply Finset.sum_congr rfl |
| intro b _ |
| dsimp only [transportedMagnetization] |
| field_simp [(hδ b).ne'] |
| _ = ∑ b, ∑ a, w a b*c b*mag a := by |
| simp_rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro b _ |
| apply Finset.sum_congr rfl |
| intro a _ |
| ring |
| _ = _ := Finset.sum_comm |
|
|
| lemma transportedMagnetization_field_cost {A B : Type*} [Fintype A] [Fintype B] |
| (w : A → B → ℝ) (γ b : A → ℝ) (δ c : B → ℝ) (mag : A → ℝ) |
| (hw : ∀ a j, 0 ≤ w a j) (hδ : ∀ j, 0 < δ j) |
| (hrow : ∀ a, ∑ j, w a j=γ a) (hm : ∀ a, |mag a| ≤ 1) : |
| (∑ a, γ a*b a*mag a)-(∑ j, δ j*c j*transportedMagnetization w δ mag j) ≤ |
| ∑ a, ∑ j, w a j* |b a-c j| := by |
| have hsource : (∑ a, γ a*b a*mag a)=∑ a, ∑ j, w a j*b a*mag a := by |
| simp_rw [← Finset.sum_mul,hrow] |
| rw [hsource,transportedMagnetization_field_term w δ c mag hδ,← Finset.sum_sub_distrib] |
| apply Finset.sum_le_sum |
| intro a _ |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_le_sum |
| intro j _ |
| calc |
| _ = w a j*(b a-c j)*mag a := by ring |
| _ ≤ |w a j*(b a-c j)*mag a| := le_abs_self _ |
| _ = w a j* |b a-c j| * |mag a| := by rw [abs_mul,abs_mul,abs_of_nonneg (hw a j)] |
| _ ≤ _ := mul_le_of_le_one_right (mul_nonneg (hw a j) (abs_nonneg _)) (hm a) |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
