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a89d282 | 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 | 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
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