| import OAI.Geometry.PinnedDistances.EmpiricalCompactness |
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| namespace OAI |
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| universe u1 u2 u3 u4 u5 u6 u7 |
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| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology ENNReal |
| namespace WeakPinned.Bounded |
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| variable {I : Type u1} {V : Type u2} {X : Type u3} {Y : Type u4} [Fintype I] [Fintype V] |
| [MeasurableSpace X] [MeasurableSpace Y] |
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| lemma finiteLaw_fst (w : I → ℝ) (p : I → V → ℝ) (U : I → X) |
| (hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) : |
| finiteLaw (edgeWeight w p) (fun r => U r.1)=finiteLaw w U := by |
| unfold finiteLaw edgeWeight |
| rw [Fintype.sum_prod_type] |
| apply Finset.sum_congr rfl |
| intro i _ |
| dsimp only |
| rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun x _ => mul_nonneg (hw i) (hp i x)), |
| ← Finset.mul_sum,hpsum,mul_one] |
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| lemma tripleLaw_edge (w : I → ℝ) (p : I → V → ℝ) (U : I × V → X) |
| (hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) : |
| finiteLaw (tripleWeight w p) (fun r => U (r.1,r.2.1))=finiteLaw (edgeWeight w p) U := by |
| unfold finiteLaw tripleWeight edgeWeight |
| simp only [Fintype.sum_prod_type] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [show (∑ z,∑ t, ENNReal.ofReal (w i*p i x*p i z*p i t) • Measure.dirac (U (i,x))) = |
| ∑ z,ENNReal.ofReal (w i*p i x*p i z) • Measure.dirac (U (i,x)) by |
| apply Finset.sum_congr rfl |
| intro z _ |
| rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun t _ => mul_nonneg (mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)) (hp i t)), |
| ← Finset.mul_sum,hpsum,mul_one]] |
| rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun z _ => mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)), |
| ← Finset.mul_sum,hpsum,mul_one] |
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| lemma finiteLaw_mono (w v : I → ℝ) (U : I → X) (hle : ∀ i,w i≤v i) : |
| finiteLaw w U≤finiteLaw v U := by |
| apply Measure.le_iff.mpr |
| intro s hs |
| simp only [finiteLaw,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] |
| apply Finset.sum_le_sum |
| intro i _ |
| gcongr |
| exact hle i |
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| lemma finiteLaw_prod (w : I → ℝ) (v : V → ℝ) (U : I → X) (W : V → Y) |
| (hw : ∀ i,0≤w i) : |
| (finiteLaw w U).prod (finiteLaw v W)= |
| finiteLaw (fun r : I × V => w r.1*v r.2) (fun r => (U r.1,W r.2)) := by |
| simp only [finiteLaw,← Measure.sum_fintype,Measure.prod_sum, |
| Measure.prod_smul_left,Measure.prod_smul_right,Measure.dirac_prod_dirac, |
| smul_smul,ENNReal.ofReal_mul (hw _),mul_comm] |
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| lemma finiteLaw_smul (w : I → ℝ) (U : I → X) (c : ℝ) (hc : 0≤c) : |
| finiteLaw (fun i => c*w i) U=ENNReal.ofReal c •finiteLaw w U := by |
| simp only [finiteLaw,ENNReal.ofReal_mul hc,Finset.smul_sum,smul_smul] |
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| end WeakPinned.Bounded |
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| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology ENNReal |
| namespace WeakPinned.Bounded |
| open Variance |
| variable {V : Type u5} {I : Type u6} {X : Type u7} [Fintype V] [DecidableEq V] [Fintype I] |
| [MeasurableSpace X] [MeasurableSingletonClass X] |
| [TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X] |
| variable (G : Core.Fibers V I) |
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| lemma graph_edge_eval (F : V → V → ℝ) : |
| (∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*F (G.source e) y)= |
| ∑ x,∑ y,G.edgeLaw x y*F x y := by |
| rw [← G.incoming_eval] |
| simp only [Finset.mul_sum,← mul_assoc,G.sampling_conditional] |
| exact Finset.sum_comm |
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| lemma graph_edge_law (U : V → V → X) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) : |
| finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (fun r => U (G.source r.1) r.2)= |
| finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2) := by |
| have hw (r : I × V) : 0≤edgeWeight G.fiberWeight (fun e => prob (G.targets e)) r := |
| mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _) |
| let : IsProbabilityMeasure (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (fun r => U (G.source r.1) r.2)) := finiteLaw_probability _ _ hw |
| (sum_edgeWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e))) |
| let : IsProbabilityMeasure (finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2)) := |
| finiteLaw_probability _ _ (fun r => G.edgeLaw_nonneg _ _) |
| (by simpa only [Fintype.sum_prod_type] using G.sum_edgeLaw hG) |
| apply ext_of_forall_integral_eq_of_IsFiniteMeasure |
| intro f |
| rw [integral_finiteLaw _ _ hw,integral_finiteLaw _ _ (fun r => G.edgeLaw_nonneg _ _)] |
| simp only [edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum] |
| exact graph_edge_eval G (fun x y => f (U x y)) |
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| omit G in |
| lemma pairProb_le_uniform (hn : 2≤Fintype.card V) (x y : V) : |
| pairProb Finset.univ x y≤2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹ := by |
| have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast hn |
| have hn0 : (0:ℝ)<Fintype.card V := by linarith |
| simp only [pairProb,Finset.mem_univ,true_and,Finset.card_univ] |
| split_ifs |
| · field_simp |
| apply (div_le_iff₀ (by linarith : 0<(Fintype.card V:ℝ)-1)).mpr |
| nlinarith |
| · positivity |
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| omit [Fintype I] [MeasurableSingletonClass X] [TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X] in |
| lemma graph_domination (U : V → X) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M) |
| (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) : |
| finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => (U r.1,U r.2))≤ |
| ENNReal.ofReal (2*M) • (uniformLaw U).prod (uniformLaw U) := by |
| rw [uniformLaw,finiteLaw_prod _ _ _ _ (fun _ => by positivity),← finiteLaw_smul _ _ _ (by positivity)] |
| apply finiteLaw_mono |
| intro r |
| calc |
| _ ≤ M*pairProb Finset.univ r.1 r.2 := hdom _ _ |
| _ ≤ M*(2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹) := |
| mul_le_mul_of_nonneg_left (pairProb_le_uniform hn _ _) hM |
| _ = _ := by ring |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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