| import Mathlib |
| import OAI.Analysis.FreeWords.AmplifiedBlockModelDefinitions |
| import OAI.Analysis.FreeWords.AmplifiedCyclicApprox |
| import OAI.Analysis.FreeWords.CyclicApproxNet |
| import OAI.Analysis.FreeWords.MatrixPowerBlockLimits |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKFreeLiftFrontier |
|
|
| open KKTraceTransfer KKMatrixExpectation KKHaarEndpoints Filter |
|
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| open scoped Topology |
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| noncomputable section |
|
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| theorem finite_amplified_block_model : FiniteAmplifiedBlockModel (H := H) := by |
| classical |
| intro M τ hfac hdiff n hn K F E hF hE ε hε |
| let τn := matrixTraceMap τ.map n |
| let B := (numericalHom (A := ConcreteAlgebra M) n).range.toSubalgebra |
| obtain ⟨u,hgood,hH,hC⟩ := cyclic_approx_net τn B (FullMatrixUnitary n) |
| (finite_amplified_cyclic_approx M τ hfac hdiff hn) |
| have hblocks := matrix_power_block_limits (A := ConcreteAlgebra M) |
| (CyclicIndex τn B) atTop τ.map τ.normalized τ.tracial n hn u hH hC |
| have hKsmall : ∀ᶠj in atTop,∀k∈K,k≠0 → ‖τn (u j^k).val‖<ε := by |
| apply (Filter.eventually_all_finset K).2 |
| intro k hk |
| by_cases hk0 : k=0 |
| · exact Eventually.of_forall (fun _ h => (h hk0).elim) |
| · have hh := Metric.tendsto_nhds.mp (hH k hk0) ε hε |
| simpa only [dist_zero_right] using hh.mono (fun j hj _ => hj) |
| have hFsmall : ∀ᶠj in atTop,∀w∈F,‖τn (powerWord (u j) w)‖<ε := by |
| apply (Filter.eventually_all_finset F).2 |
| intro w hw |
| simpa only [dist_zero_right] using Metric.tendsto_nhds.mp (hC w (hF w hw).1 (hF w hw).2) ε hε |
| have hEsmall : ∀ᶠj in atTop,∀z∈E,∀a b, |
| ‖τ.map ((powerWord (u j) z.1*(u j^z.2).val) a b)‖<ε := by |
| apply (Filter.eventually_all_finset E).2 |
| intro z hz |
| apply Filter.eventually_all.2 |
| intro a |
| apply Filter.eventually_all.2 |
| intro b |
| simpa only [dist_zero_right] using Metric.tendsto_nhds.mp |
| (hblocks z.1 (hE z hz).1 z.2 (hE z hz).2 a b) ε hε |
| obtain ⟨j,hjK,hjF,hjE⟩ := (hKsmall.and (hFsmall.and hEsmall)).exists |
| obtain ⟨d,hd,ρ,hρ,hU⟩ := hgood j |
| exact ⟨d,hd,ρ,hρ,u j,hU,hjK,hjF,hjE⟩ |
|
|
| end |
|
|
| end KKFreeLiftFrontier |
| end |
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| end OAI |
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|
