| 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 |
|
|
