File size: 3,064 Bytes
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 | 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
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