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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 73 74 75 | 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
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