| import OAI.Probability.InvariantIsing.Core.FiniteFieldMatching |
|
|
| namespace OAI |
|
|
| /-! The unmatched field sites are controlled by differences of label counts. -/ |
| noncomputable section |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma fieldLabelCount_sum {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] |
| (g : Fin N → A) : ∑ a, spinGroupSize g a=N := by |
| simpa only [Fintype.card_sigma, Fintype.card_subtype, spinGroupSize, Fintype.card_fin] using |
| Fintype.card_congr (Equiv.sigmaFiberEquiv g) |
|
|
| theorem exists_field_permutation_l1_bound {N : ℕ} {A : Type*} [Fintype A] [DecidableEq A] |
| (g h : Fin N → A) (b : A → ℝ) {B : ℝ} (hB : 0 ≤ B) (hb : ∀ a, |b a| ≤ B) : |
| ∃ p : Equiv.Perm (Fin N), |
| (∑ i, |b (g i)-b (h (p i))|) ≤ |
| 2*B*∑ a, |(spinGroupSize g a : ℝ)-spinGroupSize h a| := by |
| obtain ⟨p,hp⟩ := exists_field_label_matching g h |
| let S := Finset.univ.filter (fun i => g i=h (p i)) |
| let T := Finset.univ \ S |
| have hsmall : ∑ a, min (spinGroupSize g a) (spinGroupSize h a) ≤ N := |
| (Finset.sum_le_sum (fun a _ => min_le_left _ _)).trans_eq (fieldLabelCount_sum g) |
| have hmiss : ((N-∑ a, min (spinGroupSize g a) (spinGroupSize h a) : ℕ) : ℝ) ≤ |
| ∑ a, |(spinGroupSize g a : ℝ)-spinGroupSize h a| := by |
| rw [Nat.cast_sub hsmall] |
| have hn : (N : ℝ)=∑ a, (spinGroupSize g a : ℝ) := by |
| exact_mod_cast (fieldLabelCount_sum g).symm |
| rw [hn, Nat.cast_sum, ← Finset.sum_sub_distrib] |
| apply Finset.sum_le_sum |
| intro a _ |
| by_cases ha : spinGroupSize g a ≤ spinGroupSize h a |
| · rw [min_eq_left ha, sub_self] |
| exact abs_nonneg _ |
| · rw [min_eq_right (by omega)] |
| exact le_abs_self _ |
| have hT : T.card ≤ N-∑ a, min (spinGroupSize g a) (spinGroupSize h a) := by |
| rw [Finset.card_sdiff_of_subset (Finset.subset_univ S)] |
| simpa only [Finset.card_univ, Fintype.card_fin] using Nat.sub_le_sub_left hp N |
| have hsum : (∑ i, |b (g i)-b (h (p i))|) ≤ (T.card : ℝ)*(2*B) := by |
| have he : (∑ i, |b (g i)-b (h (p i))|)=∑ i ∈ T, |b (g i)-b (h (p i))| := by |
| symm |
| apply Finset.sum_subset (Finset.subset_univ T) |
| intro i _ hi |
| have hg : g i=h (p i) := by |
| by_contra hn |
| exact hi (Finset.mem_sdiff.mpr ⟨Finset.mem_univ _,by simpa only [S,Finset.mem_filter,Finset.mem_univ,true_and] using hn⟩) |
| rw [hg,sub_self,abs_zero] |
| rw [he] |
| calc |
| _ ≤ ∑ _i ∈ T, 2*B := Finset.sum_le_sum fun i _ => |
| (abs_sub _ _).trans (by linarith [hb (g i),hb (h (p i))]) |
| _ = _ := by simp |
| refine ⟨p, hsum.trans ?_⟩ |
| calc |
| (T.card : ℝ)*(2*B) ≤ ((N-∑ a, min (spinGroupSize g a) (spinGroupSize h a) : ℕ) : ℝ)*(2*B) := |
| mul_le_mul_of_nonneg_right (Nat.cast_le.mpr hT) (by positivity) |
| _ ≤ (∑ a, |(spinGroupSize g a : ℝ)-spinGroupSize h a|)*(2*B) := |
| mul_le_mul_of_nonneg_right hmiss (by positivity) |
| _ = _ := mul_comm _ _ |
|
|
| end InvariantIsing |
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| end |
|
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| end OAI |
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