| import OAI.Probability.InvariantIsing.Core.FiniteFieldAtomMass |
|
|
| namespace OAI |
|
|
| /-! Two positive finite representations of the same field law have a |
| coupling supported on identical field values. -/ |
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def sameFieldCoupling {A B : Type*} [Fintype A] |
| (γ c : A → ℝ) (δ d : B → ℝ) (a : A) (j : B) : ℝ := |
| if c a=d j then γ a*δ j/fieldAtomMass γ c (c a) else 0 |
|
|
| lemma sameFieldCoupling_nonneg {A B : Type*} [Fintype A] |
| (γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ a, 0 < γ a) (hδ : ∀ j, 0 ≤ δ j) (a : A) (j : B) : |
| 0 ≤ sameFieldCoupling γ c δ d a j := by |
| unfold sameFieldCoupling |
| split_ifs |
| · exact div_nonneg (mul_nonneg (hγ a).le (hδ j)) (fieldAtomMass_pos_at γ c hγ a).le |
| · rfl |
|
|
| lemma sameFieldCoupling_row {A B : Type*} [Fintype A] [Fintype B] |
| (γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ a, 0 < γ a) |
| (he : ∀ x, fieldAtomMass γ c x=fieldAtomMass δ d x) (a : A) : |
| ∑ j, sameFieldCoupling γ c δ d a j=γ a := by |
| have hw j : sameFieldCoupling γ c δ d a j= |
| (γ a/fieldAtomMass γ c (c a))*(if d j=c a then δ j else 0) := by |
| unfold sameFieldCoupling |
| by_cases hh : c a=d j |
| · simp only [hh,eq_self,ite_true] |
| ring |
| · have hh' : ¬ d j=c a := Ne.symm hh |
| simp only [hh,hh',ite_false,mul_zero] |
| simp_rw [hw] |
| rw [← Finset.mul_sum,show (∑ j, if d j=c a then δ j else 0)=fieldAtomMass δ d (c a) from rfl, |
| ← he (c a)] |
| exact div_mul_cancel₀ _ (fieldAtomMass_pos_at γ c hγ a).ne' |
|
|
| lemma sameFieldCoupling_col {A B : Type*} [Fintype A] [Fintype B] |
| (γ c : A → ℝ) (δ d : B → ℝ) (hδ : ∀ j, 0 < δ j) |
| (he : ∀ x, fieldAtomMass γ c x=fieldAtomMass δ d x) (j : B) : |
| ∑ a, sameFieldCoupling γ c δ d a j=δ j := by |
| have hw a : sameFieldCoupling γ c δ d a j= |
| (δ j/fieldAtomMass γ c (d j))*(if c a=d j then γ a else 0) := by |
| unfold sameFieldCoupling |
| by_cases hh : c a=d j |
| · simp only [hh,eq_self,ite_true] |
| ring |
| · simp only [hh,ite_false,mul_zero] |
| simp_rw [hw] |
| rw [← Finset.mul_sum,show (∑ a, if c a=d j then γ a else 0)=fieldAtomMass γ c (d j) from rfl] |
| apply div_mul_cancel₀ |
| rw [he] |
| exact (fieldAtomMass_pos_at δ d hδ j).ne' |
|
|
| lemma sameFieldCoupling_cost {A B : Type*} [Fintype A] [Fintype B] |
| (γ c : A → ℝ) (δ d : B → ℝ) : |
| (∑ a, ∑ j, sameFieldCoupling γ c δ d a j* |c a-d j|)=0 := by |
| apply Finset.sum_eq_zero |
| intro a _ |
| apply Finset.sum_eq_zero |
| intro j _ |
| unfold sameFieldCoupling |
| split_ifs with h |
| · simp only [h,sub_self,abs_zero,mul_zero] |
| · exact zero_mul _ |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
