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2fca478 | 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 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Geometry.PinnedDistances.BoundedRates
namespace OAI
universe u1 u2 u3
section
noncomputable section
open Filter MeasureTheory
open scoped Topology BigOperators
namespace WeakPinned
open Bounded Global Variance Hierarchy NumberField
variable {V : ℕ → Type u1} {I : ℕ → Type u2} {K : ℕ → Type u3} [∀ j,Fintype (V j)] [∀ j,Fintype (I j)] [∀ j,DecidableEq (V j)]
[∀ j,Field (K j)] [∀ j,NumberField (K j)]
variable (G : ∀ j,Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
(Z₁ Z₂ : ∀ j,V j → K j) (h₁ : ∀ j,Function.Injective (Z₁ j)) (h₂ : ∀ j,Function.Injective (Z₂ j))
/-- The bounded arithmetic regime is impossible as well. The embedding and
normalization are chosen from the actual global mass and finite graph laws. -/
theorem arithmetic_no_bounded (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hc : 0<c) (hcγ : c<γ)
(hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
(hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j,0≤η j) (hηb : ∀ j,η j≤θ/2)
(hG : ∀ j,PreparedGraph (V j) (I j) (G j) θ γ c (η j))
(t : ∀ j,I j → K j) (ht : ∀ j e y,y∈(G j).targets e →
(Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e)
(C : ℝ) (hC : 0≤C) (hscale : ∀ j,overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+
potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)≤C) : False := by
let n := fun j => (Fintype.card (V j):ℝ)
let T := fun j => n j^(1-c)
let S := fun j => n j^γ
let q := fun j => varianceCell (G j).fiberWeight (G j).targets (T j)
let r := fun j => varianceCell ((G j).retainedWeight (S j)) ((G j).retainedSet (S j)) (T j)
have hq (j D) : 0≤q j D := varianceCell_nonneg _ _ _ (G j).fiberWeight_nonneg D
have hr (j D) : 0≤r j D := varianceCell_nonneg _ _ _ ((G j).retainedWeight_nonneg _) D
have h2 (j) : 2≤Fintype.card (V j) := (hG j).card_ge
have hn2 (j) : (2:ℝ)≤n j := by dsimp only [n]; exact_mod_cast h2 j
have hn0 (j) : 0<n j := by linarith [hn2 j]
have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (by linarith [hηb j])
have hdom (j x y) := (hG j).uniform_domination hθ (hηb j) x y
have hT (j) : 1≤T j := Real.one_le_rpow (by linarith [hn2 j]) (by linarith)
have hS (j) : 1<S j := Real.one_lt_rpow (by linarith [hn2 j]) hγ
have hSn (j) : S j≤n j := by
simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le (show 1≤n j by linarith [hn2 j]) hγ1.le
have hB (j e) : 2≤((G j).targets e).card := by
have hh := (hS j).trans ((hG j).fiber e)
exact_mod_cast hh
have hR (j y) : 2≤((G j).retainedSet (S j) y).card := by
have hh := (hS j).trans_le ((G j).retainedSet_size (S j) (hSn j) y)
exact_mod_cast hh
have hvar := sampling_variance_rates G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
let W := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
have hW (j) : 0≤W j := add_nonneg (overlapScale_nonneg _ _ _ _) (potentialScale_nonneg _ _ _ _)
have hf0 (j) : 0≤fiberVariance G c Z₁ Z₂ h₁ h₂ j :=
Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ D) (hq j D))
have hr0 (j) : 0≤ incomingVariance G γ c Z₁ Z₂ h₁ h₂ j :=
Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ D) (hr j D))
have hft := bounded_denominator_rate _ W C hf0 hW hscale hvar.1
have hrt := bounded_denominator_rate _ W C hr0 hW hscale hvar.2
obtain ⟨v,hv,hvf,hvr⟩ := bounded_place_selection Z₁ Z₂ h₁ h₂ h2 q r hq hr C hC hscale hft hrt
let z₁ := fun j x => (v j).embedding (Z₁ j x)
let z₂ := fun j x => (v j).embedding (Z₂ j x)
have hz₁ (j) : Function.Injective (z₁ j) := (v j).embedding.injective.comp (h₁ j)
have hz₂ (j) : Function.Injective (z₂ j) := (v j).embedding.injective.comp (h₂ j)
let u₁ := fun j => logNormalize (z₁ j)
let u₂ := fun j => logNormalize (z₂ j)
let D : ℝ := 4*(2*C+1)+8
have hD : 0≤D := by dsimp [D]; positivity
have hlog₁ (j) : logAverage (u₁ j)≤D := by
rw [logNormalize_bound _ (hz₁ j) (h2 j)]
exact (arch_log_center_bound (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) (h2 j) (v j)).trans (by dsimp [D]; linarith [hv j])
have hlog₂ (j) : logAverage (u₂ j)≤D := by
rw [logNormalize_bound _ (hz₂ j) (h2 j)]
have hh := arch_log_center_bound (Z₂ j) (Z₁ j) (h₂ j) (h₁ j) (h2 j) (v j)
rw [archMean_scale_swap] at hh
exact hh.trans (by dsimp [D]; linarith [hv j])
choose x₁ hx₁ using (fun j => log_center (u₁ j) D (h2 j) (hlog₁ j))
choose x₂ hx₂ using (fun j => log_center (u₂ j) D (h2 j) (hlog₂ j))
have hsmall := power_fraction_rate n hn0 hn c hc
let m₁ := fun j => pairMean (rawDepth (z₁ j))
let m₂ := fun j => pairMean (rawDepth (z₂ j))
have hd₁ := sphere_arch_test_rate Z₁ Z₂ h₁ h₂ v m₁ (fun j => -u₁ j (x₁ j))
(fun j => (G j).fiberWeight) (fun j => (G j).targets) T (2/θ) h2 hT (by positivity)
(fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) hB
(fun j => (G j).fiber_domination _ (hdom j) (h2 j)) hvf hsmall
have hvf₂ : Tendsto (fun j => archMean (Z₂ j) (Z₁ j) (massEval (q j)) (v j)) atTop (𝓝 0) := by
simpa only [archMean_mass_swap] using hvf
have hd₂ := sphere_arch_test_rate Z₂ Z₁ h₂ h₁ v m₂ (fun j => -u₂ j (x₂ j))
(fun j => (G j).fiberWeight) (fun j => (G j).targets) T (2/θ) h2 hT (by positivity)
(fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) hB
(fun j => (G j).fiber_domination _ (hdom j) (h2 j)) hvf₂ hsmall
have hdR := sphere_arch_test_rate Z₁ Z₂ h₁ h₂ v m₁ (fun j => -u₁ j (x₁ j))
(fun j => (G j).retainedWeight (S j)) (fun j => (G j).retainedSet (S j)) T (2/θ) h2 hT (by positivity)
(fun j => (G j).retainedWeight_nonneg _) (fun j => (G j).sum_retainedWeight _ (hpos j)) hR
(fun j => (G j).retained_domination _ _ (hdom j) (h2 j)) hvr hsmall
apply normalized_graph_impossible G u₁ u₂ x₁ x₂
(fun j => logNormalize_injective _ (hz₁ j)) (fun j => logNormalize_injective _ (hz₂ j))
hpos (fun j e => Finset.card_pos.mp (by have := hB j e; omega)) h2 hn D hD hlog₁ hlog₂ hx₁ hx₂
(by simpa only [sub_eq_add_neg,u₁,logNormalize,z₁,m₁] using hd₁)
(by simpa only [sub_eq_add_neg,u₂,logNormalize,z₂,m₂] using hd₂)
S (fun j => Real.rpow_pos_of_pos (hn0 j) γ)
(by simpa only [sub_eq_add_neg,u₁,logNormalize,z₁,m₁] using hdR)
(omitted_rate G θ γ c η hθ hγ1 hηb hn hG) (2/θ) (by positivity) hdom
intro j e y z hy hz
have he := (ht j e y hy).trans (ht j e z hz).symm
have hem := congrArg (v j).embedding he
simp only [map_mul,map_sub] at hem
dsimp only [u₁,u₂,logNormalize,z₁,z₂]
linear_combination (Real.exp (pairMean (rawDepth (z₁ j))):ℂ)*
(Real.exp (pairMean (rawDepth (z₂ j))):ℂ)*hem
end WeakPinned
end
end
end OAI
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