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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 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Probability.InvariantIsing.Cavity.CavitySelectedGroupFrames
namespace OAI
/-! Grouping retained and exceptional base axes by the same spectral
labels. Their dimensions are the retained counts plus the finite extras. -/
noncomputable section
open scoped Matrix
namespace InvariantIsing
def cavityBaseGroupSumEquiv {m d : ℕ} (k : Fin m → ℕ) (g : Fin d → Fin m) :
((a : Fin m) × (Fin (k a) ⊕ {j : Fin d // g j=a})) ≃
(((a : Fin m) × Fin (k a)) ⊕ Fin d) where
toFun x := match x with
| ⟨a,Sum.inl i⟩ => Sum.inl ⟨a,i⟩
| ⟨_,Sum.inr j⟩ => Sum.inr j.1
invFun x := match x with
| Sum.inl ⟨a,i⟩ => ⟨a,Sum.inl i⟩
| Sum.inr j => ⟨g j,Sum.inr ⟨j,rfl⟩⟩
left_inv x := by
rcases x with ⟨a,i | ⟨j,hj⟩⟩
· rfl
· cases hj
rfl
right_inv x := by
rcases x with ⟨a,i⟩ | j <;> rfl
lemma cavityBaseGroupSumEquiv_label {m d : ℕ} (k : Fin m → ℕ) (g : Fin d → Fin m)
(x : (a : Fin m) × (Fin (k a) ⊕ {j : Fin d // g j=a})) :
Sum.elim (fun w => w.1) g (cavityBaseGroupSumEquiv k g x)=x.1 := by
rcases x with ⟨a,i | j⟩
· rfl
· exact j.2
def cavityBaseGroupDimension {m d : ℕ} (k : Fin m → ℕ) (g : Fin d → Fin m)
(a : Fin m) : ℕ := k a+Fintype.card {j : Fin d // g j=a}
def cavityBaseGroupEquiv {N m d : ℕ} (k : Fin m → ℕ)
(e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) (g : Fin d → Fin m) :
((a : Fin m) × Fin (cavityBaseGroupDimension k g a)) ≃ Fin N :=
(Equiv.sigmaCongrRight (fun a => finSumFinEquiv.symm.trans
(Equiv.sumCongr (Equiv.refl _) (Fintype.equivFin {j : Fin d // g j=a}).symm))).trans
((cavityBaseGroupSumEquiv k g).trans e)
lemma cavityBaseGroupEquiv_label {N m d : ℕ} (k : Fin m → ℕ)
(e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) (g : Fin d → Fin m)
(i : Fin N) :
((cavityBaseGroupEquiv k e g).symm i).1 = Sum.elim (fun w => w.1) g (e.symm i) := by
let E := cavityBaseGroupEquiv k e g
have h (x : (a : Fin m) × Fin (cavityBaseGroupDimension k g a)) :
Sum.elim (fun w => w.1) g (e.symm (E x))=x.1 := by
change Sum.elim (fun w => w.1) g (e.symm (e (cavityBaseGroupSumEquiv k g _)))=x.1
rw [e.symm_apply_apply, cavityBaseGroupSumEquiv_label]
rfl
simpa only [E, Equiv.apply_symm_apply] using (h (E.symm i)).symm
lemma cavityBaseGroupDimension_ge {m d : ℕ} (k : Fin m → ℕ) (g : Fin d → Fin m)
(a : Fin m) : k a ≤ cavityBaseGroupDimension k g a := Nat.le_add_right _ _
theorem cavity_canonical_group_frames {N m d : ℕ} (k : Fin m → ℕ)
(e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) (g : Fin d → Fin m)
(hk : ∀ a, d ≤ k a) :
∃ A : (a : Fin m) → Matrix (Fin (cavityBaseGroupDimension k g a)) (Fin d) ℝ,
(∀ a, (A a).transpose*A a=1) ∧
cavityGroupedSelectedFrame (cavityBaseGroupDimension k g) (cavityBaseGroupEquiv k e g)
(fun j => (g j,j)) A = cavityCanonicalSpecial e := by
have hlabel j : ((cavityBaseGroupEquiv k e g).symm (e (Sum.inr j))).1=g j := by
rw [cavityBaseGroupEquiv_label, e.symm_apply_apply]
rfl
obtain ⟨f,hf⟩ := cavity_group_selected_embedding (cavityBaseGroupDimension k g)
(cavityBaseGroupEquiv k e g) (fun j => e (Sum.inr j))
(e.injective.comp Sum.inr_injective) g hlabel
(fun a => (hk a).trans (cavityBaseGroupDimension_ge k g a))
refine ⟨cavitySelectedGroupFrame (cavityBaseGroupDimension k g) f,
cavitySelectedGroupFrame_gram _ f, ?_⟩
rw [cavitySelectedGroupFrame_columns]
ext i j
rw [hf]
rfl
end InvariantIsing
end
end OAI
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