| import OAI.Analysis.C0Absorption.FiniteShift |
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| namespace OAI |
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| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
| open NormedSpace |
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| namespace C0Absorption |
| noncomputable section |
| open Set Filter Topology NormedSpace |
| open scoped NNReal BigOperators ZeroAtInfty |
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| theorem mixed_vector_tendsto (s : C0Ball) (lev : Level) (b : Block lev) : |
| Tendsto (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).vector lev b) |
| (nhds (s,s)) (nhds ((stateWeights s).vector lev b)) := by |
| classical |
| have h1 : Tendsto (fun p : C0Ball × C0Ball => blockVector lev p.1 b) (nhds (s,s)) (nhds (blockVector lev s b)) := |
| ((blockVector_continuous lev b).comp continuous_fst).continuousAt |
| have h2 : Tendsto (fun p : C0Ball × C0Ball => blockVector lev p.2 b) (nhds (s,s)) (nhds (blockVector lev s b)) := |
| ((blockVector_continuous lev b).comp continuous_snd).continuousAt |
| have he : (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).vector lev b)= |
| (fun p : C0Ball × C0Ball => if localRadius lev p.2.val≤localRadius lev p.1.val then blockVector lev p.1 b else blockVector lev p.2 b) := by |
| funext p |
| dsimp only [mixedWeights,mixedState] |
| split_ifs <;> rfl |
| rw [he] |
| exact h1.if' h2 |
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| theorem mixed_row_tendsto (s : C0Ball) (lev : Level) (b : Block lev) : |
| Tendsto (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).row lev b) |
| (nhds (s,s)) (nhds ((stateWeights s).row lev b)) := by |
| classical |
| have h1 : Tendsto (fun p : C0Ball × C0Ball => blockRow lev p.1 b) (nhds (s,s)) (nhds (blockRow lev s b)) := |
| ((blockRow_continuous lev b).comp continuous_fst).continuousAt |
| have h2 : Tendsto (fun p : C0Ball × C0Ball => blockRow lev p.2 b) (nhds (s,s)) (nhds (blockRow lev s b)) := |
| ((blockRow_continuous lev b).comp continuous_snd).continuousAt |
| have he : (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).row lev b)= |
| (fun p : C0Ball × C0Ball => if localRadius lev p.2.val≤localRadius lev p.1.val then blockRow lev p.1 b else blockRow lev p.2 b) := by |
| funext p |
| dsimp only [mixedWeights,mixedState] |
| split_ifs <;> rfl |
| rw [he] |
| exact h1.if' h2 |
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| variable {N : ℕ} |
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| theorem finiteRow_mixed_tendsto (s : C0Ball) (l : FLevel N) (x : FInput N) : |
| Tendsto (fun p : C0Ball × C0Ball => finiteRow (mixedWeights p.1 p.2) l x) |
| (nhds (s,s)) (nhds (finiteRow (stateWeights s) l x)) := by |
| exact tendsto_finsetSum _ (fun b _ => (mixed_row_tendsto s _ b).mul_const _) |
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| theorem finitePrev_mixed_tendsto (s : C0Ball) (l : FLevel N) (x : FInput N) : |
| Tendsto (fun p : C0Ball × C0Ball => finitePrev (mixedWeights p.1 p.2) l x) |
| (nhds (s,s)) (nhds (finitePrev (stateWeights s) l x)) := by |
| unfold finitePrev |
| split_ifs |
| · exact tendsto_const_nhds |
| · exact finiteRow_mixed_tendsto s _ x |
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| theorem finiteFrozen_mixed_apply_tendsto (s : C0Ball) (x : FInput N) : |
| Tendsto (fun p : C0Ball × C0Ball => finiteFrozen N (mixedWeights p.1 p.2) x) |
| (nhds (s,s)) (nhds (finiteFrozen N (stateWeights s) x)) := by |
| apply tendsto_pi_nhds.mpr |
| rintro (i|⟨l,b⟩) |
| · exact finiteRow_mixed_tendsto s (i,Fin.last N) x |
| · exact tendsto_const_nhds.add ((mixed_vector_tendsto s _ b).mul |
| ((finitePrev_mixed_tendsto s l x).sub (finiteRow_mixed_tendsto s l x))) |
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| theorem finiteFrozen_mixed_tendsto (N : ℕ) (s : C0Ball) : |
| Tendsto (fun p : C0Ball × C0Ball => finiteFrozen N (mixedWeights p.1 p.2)) |
| (nhds (s,s)) (nhds (finiteFrozen N (stateWeights s))) := by |
| classical |
| exact tendsto_finiteOperator_of_columns _ _ (fun a => finiteFrozen_mixed_apply_tendsto s _) |
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| theorem normalizedBall_continuousAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {x : E} (hx : x≠0) : ContinuousAt (@normalizedBall E _ _ ) x := by |
| apply tendsto_subtype_rng.mpr |
| exact (continuousAt_id.norm.inv₀ (norm_ne_zero_iff.mpr hx)).smul continuousAt_id |
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| theorem finite_normalized_continuousAt {x : FInput N} (hx : x≠0) : |
| ContinuousAt (fun y => normalizedBall (finiteEmbed N y)) x := by |
| have he : finiteEmbed N x≠0 := fun he => hx ((finiteEmbed N).injective (he.trans (map_zero _).symm)) |
| exact (normalizedBall_continuousAt he).comp (finiteEmbed N).continuous.continuousAt |
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| theorem finite_mixed_operator_near {x : FInput N} (hx : x≠0) : |
| ∃ s : Set (FInput N), s∈nhds x ∧ ∀ y∈s, ∀ z∈s, |
| ‖finiteFrozen N (mixedWeights (normalizedBall (finiteEmbed N y)) (normalizedBall (finiteEmbed N z)))- |
| finiteFrozen N (stateWeights (normalizedBall (finiteEmbed N x)))‖ < 1/20 := by |
| exact diagonal_neighborhood (fun y => normalizedBall (finiteEmbed N y)) |
| (finite_normalized_continuousAt hx) (fun p => finiteFrozen N (mixedWeights p.1 p.2)) |
| (finiteFrozen N (stateWeights (normalizedBall (finiteEmbed N x)))) |
| (finiteFrozen_mixed_tendsto N _) (by norm_num) |
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| theorem finiteRecomputed_local_onto {x : FInput N} (hx : x≠0) : |
| Set.range (finiteRecomputed N)∈nhds (finiteRecomputed N x) := by |
| let W := stateWeights (normalizedBall (finiteEmbed N x)) |
| let S := finiteEquiv N W |
| obtain ⟨s,hs,hclose⟩ := finite_mixed_operator_near hx |
| apply range_mem_nhds_of_approx S s hs (1/10) |
| · have hh : ‖(S.symm : FInput N →L[ℝ] FInput N)‖₊≤5 := by |
| exact_mod_cast finiteEquiv_symm_norm N W |
| have hpos : 0<‖(S.symm : FInput N →L[ℝ] FInput N)‖₊ := by |
| exact S.nnnorm_symm_pos |
| rw [inv_eq_one_div,lt_div_iff₀ hpos] |
| calc (1/10 : ℝ≥0)*‖(S.symm : FInput N →L[ℝ] FInput N)‖₊ ≤ (1/10)*5 := mul_le_mul_of_nonneg_left hh (by positivity) |
| _ < 1 := by norm_num |
| · intro y hy z hz |
| let V := mixedWeights (normalizedBall (finiteEmbed N y)) (normalizedBall (finiteEmbed N z)) |
| have he : ‖finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N V (y-z)‖≤24*eta*dist y z := |
| finiteRecomputed_mixed_approx y z |
| have hd : ‖(finiteFrozen N V-finiteFrozen N W) (y-z)‖≤(1/20)*dist y z := by |
| apply ((finiteFrozen N V-finiteFrozen N W).le_opNorm (y-z)).trans |
| rw [← dist_eq_norm y z] |
| exact mul_le_mul_of_nonneg_right (hclose y hy z hz).le dist_nonneg |
| have hid : finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N W (y-z)= |
| (finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N V (y-z))+ |
| (finiteFrozen N V-finiteFrozen N W) (y-z) := by |
| rw [sub_apply] |
| abel |
| change ‖finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N W (y-z)‖≤(1/10 : ℝ≥0)*‖y-z‖ |
| rw [hid] |
| apply (norm_add_le _ _).trans |
| calc |
| _ ≤ 24*eta*dist y z+(1/20)*dist y z := add_le_add he hd |
| _ = (24*eta+1/20)*dist y z := by ring |
| _ ≤ (1/10)*dist y z := mul_le_mul_of_nonneg_right (by norm_num [eta]) dist_nonneg |
| _ = _ := by norm_num only [NNReal.coe_div,NNReal.coe_ofNat,NNReal.coe_one]; rw [dist_eq_norm] |
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| end |
| end C0Absorption |
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| end OAI |
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