| import OAI.Probability.InvariantIsing.Cavity.CavitySelectedGroupFrames |
|
|
| namespace OAI |
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|
| /-! Grouping retained and exceptional base axes by the same spectral |
| labels. Their dimensions are the retained counts plus the finite extras. -/ |
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|
| noncomputable section |
| open scoped Matrix |
|
|
| namespace InvariantIsing |
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|
| 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 |
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| end |
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| end OAI |
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|
