| import OAI.Probability.InvariantIsing.Fields.SpinGroupSlice |
| import Mathlib.Logic.Equiv.Fintype |
|
|
| namespace OAI |
|
|
| /-! Match as many sites of each field label as both finite populations contain. -/ |
| noncomputable section |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def fieldLabelInjection {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] |
| (g : Fin N → A) (s : A → ℕ) (hs : ∀ a, s a ≤ spinGroupSize g a) : |
| (Σ a, Fin (s a)) ↪ Fin N where |
| toFun x := ((Fintype.equivFin {i : Fin N // g i=x.1}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs x.1) x.2)).val |
| inj' := by |
| rintro ⟨a,i⟩ ⟨b,j⟩ he |
| have ha : g ((Fintype.equivFin {i : Fin N // g i=a}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs a) i)).val=a := |
| ((Fintype.equivFin {i : Fin N // g i=a}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs a) i)).property |
| have hb : g ((Fintype.equivFin {i : Fin N // g i=b}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs b) j)).val=b := |
| ((Fintype.equivFin {i : Fin N // g i=b}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs b) j)).property |
| have hab : a=b := ha.symm.trans ((congrArg g he).trans hb) |
| subst b |
| have hh := (Fintype.equivFin {i : Fin N // g i=a}).symm.injective (Subtype.ext he) |
| have hij : i=j := by |
| apply Fin.ext |
| exact congrArg (fun z : Fin (Fintype.card {i : Fin N // g i=a}) => z.val) hh |
| subst j |
| rfl |
|
|
| lemma fieldLabelInjection_label {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] |
| (g : Fin N → A) (s : A → ℕ) (hs : ∀ a, s a ≤ spinGroupSize g a) |
| (x : Σ a, Fin (s a)) : g (fieldLabelInjection g s hs x)=x.1 := |
| ((Fintype.equivFin {i : Fin N // g i=x.1}).symm |
| (Fin.castLE (by simpa only [spinGroupSize, Fintype.card_subtype] using hs x.1) x.2)).property |
|
|
| theorem exists_field_label_matching {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] |
| (g h : Fin N → A) : ∃ p : Equiv.Perm (Fin N), |
| (Finset.univ.filter (fun i => g i=h (p i))).card ≥ |
| ∑ a, min (spinGroupSize g a) (spinGroupSize h a) := by |
| let s := fun a => min (spinGroupSize g a) (spinGroupSize h a) |
| let f := fieldLabelInjection g s (fun a => min_le_left _ _) |
| let k := fieldLabelInjection h s (fun a => min_le_right _ _) |
| obtain ⟨p,hp⟩ := Equiv.Perm.exists_extending_pair f k f.injective k.injective |
| refine ⟨p,?_⟩ |
| have hsub : Finset.univ.image f ⊆ Finset.univ.filter (fun i => g i=h (p i)) := by |
| intro i hi |
| obtain ⟨x,_,rfl⟩ := Finset.mem_image.mp hi |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_univ _,?_⟩ |
| rw [hp] |
| exact (fieldLabelInjection_label g s _ x).trans (fieldLabelInjection_label h s _ x).symm |
| have hc := Finset.card_le_card hsub |
| rw [Finset.card_image_of_injective _ f.injective] at hc |
| simpa only [Finset.card_univ, Fintype.card_sigma, Fintype.card_fin, s] using hc |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
