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9b6e014 | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | import OAI.Probability.ClassicalON.AmplitudePartition
namespace OAI
universe uE uV uX
noncomputable section
open MeasureTheory Set
open scoped BigOperators
namespace ClassicalON
section
variable {X : Type uX} {V : Type uV} {E : Type uE} [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X]
[BorelSpace X] [Fintype E] (μ : Measure X) [IsProbabilityMeasure μ]
(T : E → X → ℝ) (left right : E → V) (b : E → ℝ)
(hT : ∀ e,Continuous (T e)) (hCov : EdgeCovarianceNonneg μ T)
(h0 : ∀ e,0≤∫ x,T e x ∂μ) (hb : ∀ e,0≤b e)
include hT hCov h0 hb
theorem log_amplitudePartition_increment_mono {a c d : V → ℝ}
(ha : ∀ v,0≤a v) (hac : ∀ v,a v≤c v) (hd : ∀ v,0≤d v) :
Real.log (amplitudePartition μ T left right b (fun v => a v+d v))-
Real.log (amplitudePartition μ T left right b a) ≤
Real.log (amplitudePartition μ T left right b (fun v => c v+d v))-
Real.log (amplitudePartition μ T left right b c) := by
let F := fun t => Real.log (amplitudePartition μ T left right b (fun v => c v+t*d v))-
Real.log (amplitudePartition μ T left right b (fun v => a v+t*d v))
have hD (t : ℝ) := (log_amplitudePartition_direction_derivative μ T left right b hT c d t).sub
(log_amplitudePartition_direction_derivative μ T left right b hT a d t)
have hM : MonotoneOn F (Icc 0 1) := by
apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 1)
(fun t _ => (hD t).continuousAt.continuousWithinAt)
(fun t _ => (hD t).hasDerivWithinAt)
intro t ht
have ht0 : 0≤t := (interior_subset ht).1
apply sub_nonneg.mpr
apply Finset.sum_le_sum
intro e _
have hJa : ∀ g,0≤amplitudeCoupling left right b (fun v => a v+t*d v) g := by
intro g
unfold amplitudeCoupling
exact mul_nonneg (mul_nonneg (hb g) (add_nonneg (ha _) (mul_nonneg ht0 (hd _))))
(add_nonneg (ha _) (mul_nonneg ht0 (hd _)))
have hJ : ∀ g,amplitudeCoupling left right b (fun v => a v+t*d v) g≤
amplitudeCoupling left right b (fun v => c v+t*d v) g := by
apply amplitudeCoupling_mono left right b hb
· intro v; exact add_nonneg (ha v) (mul_nonneg ht0 (hd v))
· intro v; exact add_le_add (hac v) le_rfl
have hm := edgeMean_mono_couplings μ T hT hCov hJa hJ e
apply mul_le_mul (amplitudeSlope_mono left right b hb hd hac t e) hm
· exact edgeMean_nonneg μ T hT hCov h0 _ hJa e
· exact amplitudeSlope_nonneg left right b hb (fun v => (ha v).trans (hac v)) hd ht0 e
have hi := hM (by simp : (0 : ℝ)∈Icc 0 1) (by simp : (1 : ℝ)∈Icc 0 1) (by norm_num)
simp only [F,zero_mul,add_zero,one_mul] at hi
linarith
theorem amplitudePartition_supermodular {u v : V → ℝ}
(hu : ∀ i,0≤u i) (hv : ∀ i,0≤v i) :
amplitudePartition μ T left right b u*amplitudePartition μ T left right b v ≤
amplitudePartition μ T left right b (u⊓v)*amplitudePartition μ T left right b (u⊔v) := by
let a := u⊓v
let d := fun i => u i-a i
have ha : ∀ i,0≤a i := by intro i; exact le_min (hu i) (hv i)
have hd : ∀ i,0≤d i := by intro i; exact sub_nonneg.mpr (min_le_left _ _)
have had : (fun i => a i+d i)=u := by funext i; dsimp [d]; ring
have hvd : (fun i => v i+d i)=u⊔v := by
funext i
dsimp [d,a]
rcases le_total (u i) (v i) with h | h
· rw [min_eq_left h,max_eq_right h]; ring
· rw [min_eq_right h,max_eq_left h]; ring
have hi := log_amplitudePartition_increment_mono μ T left right b hT hCov h0 hb
ha (fun i => min_le_right (u i) (v i)) hd
rw [had,hvd] at hi
have hp := amplitudePartition_pos μ T left right b hT
apply (Real.log_le_log_iff (mul_pos (hp u) (hp v)) (mul_pos (hp _) (hp _))).mp
rw [Real.log_mul (hp u).ne' (hp v).ne',Real.log_mul (hp _).ne' (hp _).ne']
linarith
end
end ClassicalON
end
end OAI
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