| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleScheduleEndpoint |
| import OAI.Probability.SKBarriers.Locking.NarrowScheduleMass |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedStats |
|
|
| namespace OAI |
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| section |
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|
| noncomputable section |
| open scoped BigOperators Matrix |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
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|
| @[simp] theorem narrowClockPerturbed_zero (a b c r q δ : ℝ) : |
| narrowClockPerturbed a b c r q δ 0 0 0=tripleClockMatrix a b c r q := by |
| rw [narrowClockPerturbed_pair_zero,tripleClockPerturbed_zero] |
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|
| def narrowCrossPenalty (c v w : List (ℝ × (ℝ × ℝ))) : ℝ := |
| weightedCrossPenalty c 0+weightedCrossPenalty v (weightedCross c)+weightedCrossPenalty w (weightedCross c) |
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|
| def narrowQuarticPenalty (c v w : List (ℝ × (ℝ × ℝ))) : ℝ := |
| weightedVariancePenalty c 0/3+narrowNoisePenalty (mergedProductBranches v w) (weightedVariance c) |
|
|
| theorem narrowSchedule_covariance (δ : ℝ) (c v w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) |
| (hv : weightedUnderlying v=weightedUnderlying w) (hy : weightedCross c+weightedCross w=0) : |
| matrixChainCovariance (narrowSchedule δ c v w t)= |
| tripleClockPerturbed (rawVariance (weightedUnderlying c++weightedUnderlying w++t)) |
| (rawVariance (weightedUnderlying c++weightedUnderlying w++t)) |
| (rawVariance (weightedUnderlying c++weightedUnderlying w++t)) (rawVariance (weightedUnderlying c)) |
| (rawVariance (weightedUnderlying c++weightedUnderlying w)) δ |
| (weightedCross c+weightedCross v) (narrowVariance c v w) := by |
| have hc := narrowCommonSchedule_covariance δ c 0 0 0 |
| simp only [narrowClockPerturbed_zero,tripleClockMatrix_zero,zero_add] at hc |
| have hm := narrowMiddleSchedule_covariance δ (mergedProductBranches v w) |
| (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) |
| (weightedCross c) (weightedCross c) (weightedVariance c) |
| simp only [mergedProductBranches_left,mergedProductBranches_right,hv,hy,narrowClockPerturbed_pair_zero] at hm |
| simp only [narrowSchedule,matrixChainCovariance_append,hc] |
| rw [hm,tripleTailSchedule_covariance] |
| simp only [rawVariance_append,narrowVariance] |
|
|
| theorem narrowSchedule_penalty (δ : ℝ) (c v w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) |
| (hv : weightedUnderlying v=weightedUnderlying w) (hy : weightedCross c+weightedCross w=0) : |
| matrixChainPenalty (narrowSchedule δ c v w t) 0= |
| 3*rawPenalty (weightedUnderlying c++weightedUnderlying w++t) 0+ |
| 4*δ^2*(narrowCrossPenalty c v w+narrowVariance c v w*rawArea t)+ |
| 4*δ^4*narrowQuarticPenalty c v w := by |
| have hc := narrowCommonSchedule_covariance δ c 0 0 0 |
| simp only [narrowClockPerturbed_zero,tripleClockMatrix_zero,zero_add] at hc |
| have hpc := narrowCommonSchedule_penalty δ c 0 0 0 |
| simp only [narrowClockPerturbed_zero,tripleClockMatrix_zero] at hpc |
| have hm := narrowMiddleSchedule_covariance δ (mergedProductBranches v w) |
| (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) |
| (weightedCross c) (weightedCross c) (weightedVariance c) |
| simp only [mergedProductBranches_left,mergedProductBranches_right,hv,hy,narrowClockPerturbed_pair_zero] at hm |
| have hpm := narrowMiddleSchedule_penalty δ (mergedProductBranches v w) |
| (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) (rawVariance (weightedUnderlying c)) |
| (weightedCross c) (weightedCross c) (weightedVariance c) |
| simp only [mergedProductBranches_left,mergedProductBranches_right,hv] at hpm |
| simp only [narrowSchedule,matrixChainPenalty_append,matrixChainCovariance_append,zero_add,hc,hpc,hm,hpm, |
| tripleTailSchedule_penalty,rawPenalty_append,rawVariance_append,narrowCrossPenalty,narrowVariance,narrowQuarticPenalty] |
| ring |
|
|
| theorem narrowNoisePenalty_nonneg (l : List NarrowMiddleIncrement) |
| (hm : ∀ p∈l,0≤productMass p) {K : ℝ} (hK : 0≤K) : 0≤narrowNoisePenalty l K := by |
| induction l generalizing K with |
| | nil => rfl |
| | cons p l ih => |
| change 0≤productMass p*((K+(narrowMiddleNoise p)^2)^2-K^2)+narrowNoisePenalty l (K+(narrowMiddleNoise p)^2) |
| apply add_nonneg |
| · apply mul_nonneg (hm p (List.mem_cons_self ..)) |
| nlinarith [sq_nonneg (narrowMiddleNoise p)] |
| · exact ih (fun q hq => hm q (List.mem_cons_of_mem _ hq)) (by positivity) |
|
|
| theorem narrowQuarticPenalty_nonneg (c v w : List (ℝ × (ℝ × ℝ))) |
| (hc : ∀ p∈c,0≤p.1) (hv : ∀ p∈v,0≤p.1) (hw : ∀ p∈w,0≤p.1) : |
| 0≤narrowQuarticPenalty c v w := by |
| apply add_nonneg (div_nonneg (weightedVariancePenalty_nonneg c hc le_rfl) (by norm_num)) |
| apply narrowNoisePenalty_nonneg |
| · intro p hp |
| simp only [mergedProductBranches,List.mem_merge,List.mem_map] at hp |
| rcases hp with ⟨q,hq,rfl⟩|⟨q,hq,rfl⟩ |
| · exact hv q hq |
| · exact div_nonneg (hw q hq) (by norm_num) |
| · unfold weightedVariance |
| exact List.sum_nonneg (fun x hx => by obtain ⟨p,hp,rfl⟩ := List.mem_map.mp hx; positivity) |
|
|
| theorem narrowConstrainedPressure_schedule {N : ℕ} (hN : 0<N) (β δ : ℝ) |
| (c v w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) |
| (hv : weightedUnderlying v=weightedUnderlying w) (hy : weightedCross c+weightedCross w=0) |
| (hm : ∀ p∈weightedUnderlying c++weightedUnderlying w++t,p.1∈Icc (0:ℝ) 1) |
| (hs : (weightedUnderlying c++weightedUnderlying w++t).Pairwise (fun p q => p.1≤q.1)) |
| (ht : rawVariance (weightedUnderlying c++weightedUnderlying w++t)=1) |
| (hg : weightedCross c+weightedCross v=1) |
| (hD : Nonempty (MatrixStates N 3 (tripleGram (rawVariance (weightedUnderlying c)) δ |
| (rawVariance (weightedUnderlying c++weightedUnderlying w))))) : |
| matrixConstrainedPressure N 3 β (tripleGram (rawVariance (weightedUnderlying c)) δ |
| (rawVariance (weightedUnderlying c++weightedUnderlying w))) ≤ |
| vectorIncrementChain ((narrowSchedule δ c v w t).map (fun p => (p.1,β • p.2))) |
| (fun x => ∑ u,scalarSpinTerminal (x u)) 0- |
| 3*β^2/4*rawPenalty (weightedUnderlying c++weightedUnderlying w++t) 0- |
| β^2*δ^2*(narrowCrossPenalty c v w+narrowVariance c v w*rawArea t)+ |
| β^2*δ^4*(narrowVariance c v w)^2 := by |
| have H := matrixConstrainedPressure_chain hN β _ hD (narrowSchedule δ c v w t) |
| (narrowSchedule_mass_bounds δ c v w t hv hm) |
| (narrowSchedule_monotone δ c v w t hv (fun p hp => (hm p hp).1) hs) |
| have hcw : ∀ p∈weightedUnderlying c++weightedUnderlying w,0≤p.1 := |
| fun p hp => (hm p (List.mem_append_left _ hp)).1 |
| have hc : ∀ p∈c,0≤p.1 := fun p hp => hcw (p.1,p.2.1) |
| (List.mem_append_left _ (List.mem_map.mpr ⟨p,hp,rfl⟩)) |
| have hw : ∀ p∈w,0≤p.1 := fun p hp => hcw (p.1,p.2.1) |
| (List.mem_append_right _ (List.mem_map.mpr ⟨p,hp,rfl⟩)) |
| have hv' : ∀ p∈v,0≤p.1 := by |
| intro p hp |
| have hp' : (p.1,p.2.1)∈weightedUnderlying w := hv ▸ List.mem_map.mpr ⟨p,hp,rfl⟩ |
| exact hcw _ (List.mem_append_right _ hp') |
| have hq := narrowQuarticPenalty_nonneg c v w hc hv' hw |
| simp only [narrowSchedule_covariance δ c v w t hv hy,ht,hg,tripleClockPerturbed_symmetric, |
| triplePerturbedMatrix_mismatch,narrowSchedule_penalty δ c v w t hv hy] at H |
| have hn : 0≤β^2*δ^4*narrowQuarticPenalty c v w := by positivity |
| nlinarith |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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