File size: 2,789 Bytes
c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticVariationalTransport
namespace OAI
/-! A coupling retains a common fraction of every population and sends
the remaining mass independently to the required residual population. -/
noncomputable section
open scoped BigOperators
namespace InvariantIsing
def retainedPopulationCoupling {A : Type*} [DecidableEq A]
(γ δ : A → ℝ) (ε : ℝ) (a j : A) : ℝ :=
(if a=j then (1-ε)*γ a else 0)+γ a*(δ j-(1-ε)*γ j)
lemma retainedPopulationCoupling_nonneg {A : Type*} [DecidableEq A]
(γ δ : A → ℝ) {ε : ℝ} (hε : ε ≤ 1) (hγ : ∀ a, 0 ≤ γ a)
(hδ : ∀ a, (1-ε)*γ a ≤ δ a) (a j : A) :
0 ≤ retainedPopulationCoupling γ δ ε a j := by
apply add_nonneg
· split_ifs
· exact mul_nonneg (sub_nonneg.mpr hε) (hγ a)
· rfl
· exact mul_nonneg (hγ a) (sub_nonneg.mpr (hδ j))
lemma retainedPopulationCoupling_row {A : Type*} [Fintype A] [DecidableEq A]
(γ δ : A → ℝ) (hγ : ∑ a, γ a=1) (hδ : ∑ a, δ a=1) (ε : ℝ) (a : A) :
∑ j, retainedPopulationCoupling γ δ ε a j=γ a := by
simp only [retainedPopulationCoupling,Finset.sum_add_distrib]
rw [← Finset.mul_sum,Finset.sum_sub_distrib,← Finset.mul_sum,hγ,hδ]
simp only [Finset.sum_ite_eq,Finset.mem_univ,ite_true]
ring
lemma retainedPopulationCoupling_col {A : Type*} [Fintype A] [DecidableEq A]
(γ δ : A → ℝ) (hγ : ∑ a, γ a=1) (ε : ℝ) (j : A) :
∑ a, retainedPopulationCoupling γ δ ε a j=δ j := by
simp only [retainedPopulationCoupling,Finset.sum_add_distrib]
rw [← Finset.sum_mul,hγ]
simp only [Finset.sum_ite_eq',Finset.mem_univ,ite_true]
ring
lemma retainedPopulationCoupling_cost {A : Type*} [Fintype A] [DecidableEq A]
(γ δ b : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1)
(hδsum : ∑ a, δ a=1) {ε D : ℝ}
(hδ : ∀ a, (1-ε)*γ a ≤ δ a) (hb : ∀ a, |b a| ≤ D) :
(∑ a, ∑ j, retainedPopulationCoupling γ δ ε a j* |b a-b j|) ≤ 2*D*ε := by
have he a j : retainedPopulationCoupling γ δ ε a j* |b a-b j|=
(γ a*(δ j-(1-ε)*γ j))* |b a-b j| := by
unfold retainedPopulationCoupling
split_ifs with h
· subst j; simp
· simp
simp_rw [he]
calc
_ ≤ ∑ a, ∑ j, (γ a*(δ j-(1-ε)*γ j))*(2*D) := by
apply Finset.sum_le_sum
intro a _
apply Finset.sum_le_sum
intro j _
exact mul_le_mul_of_nonneg_left ((abs_sub _ _).trans (by linarith [hb a,hb j]))
(mul_nonneg (hγ a) (sub_nonneg.mpr (hδ j)))
_ = 2*D*ε := by
simp_rw [← Finset.sum_mul,← Finset.mul_sum,Finset.sum_sub_distrib,
← Finset.mul_sum,hγsum,hδsum]
rw [← Finset.sum_mul,hγsum]
ring
end InvariantIsing
end
end OAI
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