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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 | import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure
namespace OAI
/-! Atom masses of a finitely supported external-field law, allowing
several labels to describe the same field value. -/
noncomputable section
open MeasureTheory
open scoped BigOperators
namespace InvariantIsing
def fieldAtomMass {A : Type*} [Fintype A] (γ c : A → ℝ) (x : ℝ) : ℝ :=
∑ a, if c a=x then γ a else 0
lemma finiteField_atom_mass {A : Type*} [Fintype A] (γ c : A → ℝ)
(hγ : ∀ a, 0 ≤ γ a) (x : ℝ) :
(finiteSpectralMeasure γ c).real {x}=fieldAtomMass γ c x := by
have hi := integral_finiteSpectralMeasure γ c hγ
(({x} : Set ℝ).indicator (fun _ => (1:ℝ)))
change (∫ y, ({x} : Set ℝ).indicator (1 : ℝ → ℝ) y ∂finiteSpectralMeasure γ c) = _ at hi
rw [integral_indicator_one (measurableSet_singleton x)] at hi
simpa only [fieldAtomMass,Set.indicator_apply,Set.mem_singleton_iff,mul_ite,
mul_one,mul_zero] using hi
lemma fieldAtomMass_nonneg {A : Type*} [Fintype A] (γ c : A → ℝ)
(hγ : ∀ a, 0 ≤ γ a) (x : ℝ) : 0 ≤ fieldAtomMass γ c x := by
apply Finset.sum_nonneg
intro a _
split_ifs
· exact hγ a
· rfl
lemma fieldAtomMass_pos_at {A : Type*} [Fintype A] (γ c : A → ℝ)
(hγ : ∀ a, 0 < γ a) (a : A) : 0 < fieldAtomMass γ c (c a) := by
have hh := Finset.single_le_sum (f := fun i => if c i=c a then γ i else 0)
(fun i _ => by split_ifs; exact (hγ i).le; rfl) (Finset.mem_univ a)
simp only [ite_true] at hh
exact (hγ a).trans_le hh
lemma fieldAtomMass_eq_of_law_eq {A B : Type*} [Fintype A] [Fintype B]
(γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hδ : ∀ j, 0 ≤ δ j)
(he : finiteSpectralMeasure γ c=finiteSpectralMeasure δ d) (x : ℝ) :
fieldAtomMass γ c x=fieldAtomMass δ d x := by
rw [← finiteField_atom_mass γ c hγ,← finiteField_atom_mass δ d hδ,he]
end InvariantIsing
end
end OAI
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