File size: 2,977 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 67 68 | 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
end
end OAI
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