| import OAI.Dynamics.StandardMap.ProductSum |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped ENNReal Topology Classical |
| namespace StandardMapEntropy |
| lemma BridgeGrid.scale_bound {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (k : ℝ) (hc : BridgeScalarControl k) (lo len : ℕ) (ε : ℝ) (hε : 0< ε) |
| (Fw : (Fin Rw → ℝ) → ℝ) (Fh : (Fin Rh → ℝ) → ℝ) (L : NNReal) |
| (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh) |
| (hW0 : ∀d,0≤ arrayObservation B.sw B.tw Fw d) (hH0 : ∀d,0≤ arrayObservation B.sh B.th Fh d) |
| (hlog : 1≤ Real.log (growthBase k)) (hLM : 16*Real.pi*L≤ growthBase k) |
| (hM : (2:ℝ)^10≤ growthBase k) (η : ℝ) (hη : 0<η) |
| (hδ : 1/((B.N*2^lo:ℕ):ℝ)<η/2) |
| (F : DistanceArray → ℝ) (hF : Continuous F) (D : ℝ) (hD : 0≤ D) |
| (hFW : ∀d,F d≤ D*arrayObservation B.sw B.tw Fw d) |
| (hzero : ∀d,¬((999/1000:ℝ)*((B.v:ℝ)-(B.u:ℝ))< d.val B.u B.v ∧ η< arrayObservation B.sh B.th Fh d) → F d=0) |
| (C T : ℝ) (hC : 0≤ C) |
| (hbW : ∀j<len,(∫ d,arrayObservation B.sw B.tw Fw d ∂sampleLaw k hc.nonneg (2^(B.p+lo+j)) (by positivity))≤ C*ε) |
| (hbH : ∀j<len,(∫ d,arrayObservation B.sh B.th Fh d ∂sampleLaw k hc.nonneg (2^(B.p+lo+j)) (by positivity))≤ C*ε) |
| (hT : (∫d,arrayObservation B.sw B.tw Fw d ∂scaleLaw k hc.nonneg (B.p+lo) len ε)≤ T ∨ |
| (∫d,arrayObservation B.sh B.th Fh d ∂scaleLaw k hc.nonneg (B.p+lo) len ε)≤ T) : |
| (∫d,F d ∂scaleLaw k hc.nonneg (B.p+lo) len ε)≤ |
| (D*bridgeRealConstant/(η/2))*C*(T*ε+2/((B.N*2^lo:ℕ):ℝ)) := by |
| let A (j : ℕ) := ∫d,arrayObservation B.sw B.tw Fw d ∂sampleLaw k hc.nonneg (2^(B.p+lo+j)) (by positivity) |
| let H (j : ℕ) := ∫d,arrayObservation B.sh B.th Fh d ∂sampleLaw k hc.nonneg (2^(B.p+lo+j)) (by positivity) |
| let E (j : ℕ) := growthBase k^(-(1/10:ℝ)*((B.N*2^(lo+j):ℕ):ℝ)) |
| let V (j : ℕ) := ∫d,F d ∂sampleLaw k hc.nonneg (2^(B.p+lo+j)) (by positivity) |
| have hNp : 1≤ B.N*2^lo := by |
| have hN : 0< B.N := by have := B.large; omega |
| have hh : 0< B.N*2^lo := Nat.mul_pos hN (by positivity) |
| omega |
| have hE0 (j) : 0≤ E j := by apply Real.rpow_nonneg; linarith [growthBase_ge_four k hc.nonneg] |
| have hEb (j) : E j≤1/((B.N*2^lo:ℕ):ℝ) := by |
| have hh := bridge_error_inverse (growthBase k) hM (B.N*2^lo) j hNp |
| have he : B.N*2^(lo+j)=(B.N*2^lo)*2^j := by rw [pow_add,mul_assoc] |
| dsimp [E] |
| rw [he] |
| apply hh.trans |
| apply one_div_le_one_div_of_le (by positivity) |
| exact_mod_cast Nat.le_mul_of_pos_right (B.N*2^lo) (by positivity : 0<2^j) |
| have hEs : (∑j∈Finset.range len,E j)≤2/((B.N*2^lo:ℕ):ℝ) := by |
| simpa only [E,pow_add,mul_assoc] using bridge_error_sum (growthBase k) hM (B.N*2^lo) len hNp |
| have hV (j) : V j≤ (D*bridgeRealConstant/(η/2))*((A j+E j)*H j) := by |
| have hh := B.sample_bound k hc (lo+j) Fw Fh L hFw hFh hW0 hH0 hlog hLM η hη ((hEb j).trans_lt hδ) |
| F hF D hD hFW hzero |
| dsimp only [V,A,E,H] |
| simpa only [Nat.add_assoc,mul_div_assoc,mul_assoc,div_mul_eq_mul_div] using hh |
| have hT' : (∑j∈Finset.range len,A j)≤T*ε ∨ (∑j∈Finset.range len,H j)≤T*ε := by |
| rcases hT with hT|hT |
| · left |
| rw [integral_scaleLaw k hc.nonneg (B.p+lo) len ε hε _ (continuous_arrayObservation _ _ _ hFw.continuous)] at hT |
| exact (div_le_iff₀ hε).mp hT |
| · right |
| rw [integral_scaleLaw k hc.nonneg (B.p+lo) len ε hε _ (continuous_arrayObservation _ _ _ hFh.continuous)] at hT |
| exact (div_le_iff₀ hε).mp hT |
| have hp := sum_product_error (Finset.range len) A H E C ε T hC hε.le |
| (fun j _ => integral_nonneg hW0) (fun j _ => integral_nonneg hH0) (fun j _ => hE0 j) |
| (fun j hj => hbW j (Finset.mem_range.mp hj)) (fun j hj => hbH j (Finset.mem_range.mp hj)) hT' |
| have hd : 0≤ D*bridgeRealConstant/(η/2) := div_nonneg (mul_nonneg hD bridgeRealConstant_nonneg) (by linarith) |
| rw [integral_scaleLaw k hc.nonneg (B.p+lo) len ε hε F hF] |
| apply (div_le_iff₀ hε).mpr |
| calc |
| _ ≤ ∑j∈Finset.range len,(D*bridgeRealConstant/(η/2))*((A j+E j)*H j) := Finset.sum_le_sum (fun j _ => hV j) |
| _ = (D*bridgeRealConstant/(η/2))*(∑j∈Finset.range len,(A j+E j)*H j) := by rw [Finset.mul_sum] |
| _ ≤ (D*bridgeRealConstant/(η/2))*(C*ε*(T*ε+∑j∈Finset.range len,E j)) := mul_le_mul_of_nonneg_left hp hd |
| _ ≤ (D*bridgeRealConstant/(η/2))*(C*ε*(T*ε+2/((B.N*2^lo:ℕ):ℝ))) := |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (add_le_add le_rfl hEs) (mul_nonneg hC hε.le)) hd |
| _ = _ := by ring |
| end StandardMapEntropy |
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| end OAI |
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