| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.LabelMoment |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| namespace HardSphere.Comparison |
| lemma partial_match_eq_pow {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {A : ι→Type*} [∀i,DecidableEq (A i)] (S : Finset ι) (G : ℝ) (p : ∀i,A i×A i) : |
| (∏i,if i∈S then (if (p i).1=(p i).2 then G else 1) else 1)= |
| G^(S.filter (fun i ↦ (p i).1=(p i).2)).card := by |
| have he (i : ι) : (if i∈S then (if (p i).1=(p i).2 then G else 1) else 1)= |
| if i∈S ∧ (p i).1=(p i).2 then G else 1 := by |
| by_cases hi : i∈S <;> simp [hi] |
| simp_rw [he] |
| rw [Finset.prod_ite] |
| have hs : (Finset.univ.filter (fun i:ι ↦ i∈S ∧ (p i).1=(p i).2))= |
| S.filter (fun i ↦ (p i).1=(p i).2) := by ext i; simp |
| simp only [Finset.prod_const, one_pow, mul_one, hs] |
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| lemma partial_label_match_moment {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (A : ι→Type*) [∀i,Fintype (A i)] [∀i,Nonempty (A i)] [∀i,DecidableEq (A i)] |
| [∀i,MeasurableSpace (A i)] [∀i,MeasurableSingletonClass (A i)] |
| (S : Finset ι) (G M : ℝ) (hG : 1≤G) (hM : 0<M) |
| (hcard : ∀i∈S,M≤2*(Fintype.card (A i):ℝ)) : |
| (∫p:∀i,A i×A i,G^(S.filter (fun i ↦ (p i).1=(p i).2)).card |
| ∂Measure.pi (fun i ↦ (PMF.uniformOfFintype (A i)).toMeasure.prod |
| (PMF.uniformOfFintype (A i)).toMeasure))≤ |
| Real.exp ((2*(G-1)/M)*(S.card:ℝ)) := by |
| simp_rw [←partial_match_eq_pow S G] |
| rw [integral_fintype_prod_eq_prod (fun (i:ι) (p:A i×A i) ↦ |
| if i∈S then (if p.1=p.2 then G else 1) else 1)] |
| have hi (i : ι) : (∫p:A i×A i, (if i∈S then (if p.1=p.2 then G else 1) else 1) |
| ∂((PMF.uniformOfFintype (A i)).toMeasure.prod (PMF.uniformOfFintype (A i)).toMeasure))= |
| if i∈S then 1+(G-1)/(Fintype.card (A i):ℝ) else 1 := by |
| by_cases h : i∈S |
| · simp only [h,ite_true,uniform_pair_match_integral] |
| · simp [h] |
| simp_rw [hi] |
| calc |
| _ ≤ ∏i:ι,Real.exp (if i∈S then 2*(G-1)/M else 0) := by |
| apply Finset.prod_le_prod₀ |
| · intro i _ |
| by_cases h : i∈S |
| · simp only [h,ite_true] |
| have := div_nonneg (sub_nonneg.mpr hG) (Nat.cast_nonneg (Fintype.card (A i))) |
| linarith |
| · simp [h] |
| · intro i _ |
| by_cases h : i∈S |
| · simp only [h,ite_true] |
| have hn : (0:ℝ)<Fintype.card (A i) := by exact_mod_cast Fintype.card_pos |
| have hd : (G-1)/(Fintype.card (A i):ℝ)≤2*(G-1)/M := by |
| apply (div_le_div_iff₀ hn hM).mpr |
| nlinarith [hcard i h] |
| linarith [Real.add_one_le_exp (2*(G-1)/M)] |
| · simp [h] |
| _ = _ := by |
| rw [←Real.exp_sum] |
| congr 1 |
| rw [←Finset.sum_filter] |
| simp [mul_comm] |
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| end HardSphere.Comparison |
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| end |
| end |
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| end |
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| end OAI |
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