| import OAI.Dynamics.StandardMap.RealArrays |
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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 Topology |
| open scoped ENNReal Topology CompactlySupported Classical |
| namespace StandardMapEntropy |
| lemma rescaledDistance_tree (k : ℝ) (hk : 0 ≤ k) (z : Torus) (n : ℕ) (hn : 0<n) |
| (v x y w : DyadicTime) : |
| min (rescaledDistance k z n v x+rescaledDistance k z n v y-rescaledDistance k z n x y) |
| (rescaledDistance k z n v y+rescaledDistance k z n v w-rescaledDistance k z n y w)- |
| (rescaledDistance k z n v x+rescaledDistance k z n v w-rescaledDistance k z n x w) ≤ 3/(n:ℝ) := by |
| have hh := productDistance_tree k hk z ⌊(n:ℝ)*(v:ℝ)⌋ ⌊(n:ℝ)*(x:ℝ)⌋ ⌊(n:ℝ)*(y:ℝ)⌋ ⌊(n:ℝ)*(w:ℝ)⌋ |
| have hn' : (0:ℝ)<n := by exact_mod_cast hn |
| unfold rescaledDistance |
| simp_rw [← add_div,← sub_div] |
| rw [min_div_div_right hn'.le,← sub_div] |
| exact div_le_div_of_nonneg_right (by linarith) hn'.le |
| lemma integral_nonaffine_scale_zero (k : ℝ) (hk : 0 ≤ k) (p l : ℕ) (ε : ℝ) (hε : 0<ε) |
| (g : C_c(NonAffineArray,ℝ)) |
| (hz : ∀ j∈Finset.range l,∀ z,arrayTestExtend g (sampleArray k hk z (2^(p+j)) (by positivity))=0) : |
| (∫ d,g d ∂nonaffinePart (scaleLaw k hk p l ε))=0 := by |
| rw [←integral_arrayTestExtend,integral_scaleLaw k hk p l ε hε (fun d => arrayTestExtend g d) (arrayTestExtend g).continuous] |
| have hs : (∑ j∈Finset.range l,∫ d,arrayTestExtend g d ∂sampleLaw k hk (2^(p+j)) (by positivity))=0 := by |
| apply Finset.sum_eq_zero |
| intro j hj |
| rw [integral_sampleLaw _ _ _ _ (fun d => arrayTestExtend g d) (arrayTestExtend g).continuous] |
| simp_rw [hz j hj] |
| simp |
| rw [hs,zero_div] |
| lemma integral_nonaffine_scale_zero_of_bound (k : ℝ) (hk : 0 ≤ k) (p l : ℕ) (ε : ℝ) (hε : 0<ε) |
| (g : C_c(NonAffineArray,ℝ)) (f : DistanceArray → ℝ) (q : ℝ) |
| (hg : ∀ d : NonAffineArray, f d.val ≤ q → g d=0) |
| (hf : ∀ j∈Finset.range l,∀ z,f (sampleArray k hk z (2^(p+j)) (by positivity)) ≤ q) : |
| (∫ d,g d ∂nonaffinePart (scaleLaw k hk p l ε))=0 := by |
| apply integral_nonaffine_scale_zero k hk p l ε hε g |
| intro j hj z |
| let d := sampleArray k hk z (2^(p+j)) (by positivity) |
| by_cases hd : d∈affineLocus |
| · exact arrayTestExtend_affine g d hd |
| · exact (arrayTestExtend_apply g ⟨d,hd⟩).trans (hg ⟨d,hd⟩ (hf j hj z)) |
| namespace CriticalScaleSequence |
| variable (S : CriticalScaleSequence) (L : S.LimitLaws) |
| lemma power_lower_tendsto : Tendsto (fun i => (2:ℝ)^(criticalLowerExponent S.offset i)) atTop atTop := |
| (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2)).comp S.lowerExponent_tendsto |
| lemma power_terminal_tendsto : Tendsto (fun i => (2:ℝ)^(S.exponent i)) atTop atTop := |
| (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2)).comp S.exponent_tendsto |
| lemma sample_constraint_aemulti (f : DistanceArray → ℝ) (hf : Continuous f) (C : ℝ) (hC : 0 ≤ C) |
| (hb : ∀ k : ℝ,∀ hk : 0 ≤ k,∀ z : Torus,∀ n : ℕ,∀ hn : 0<n, |
| f (sampleArray k hk z n hn) ≤ C/(n:ℝ)) : ∀ᵐ d ∂L.multi, f d.val ≤ 0 := by |
| apply ae_le_zero_of_positive_rational_null |
| intro q hq |
| have hqR : (0:ℝ)<q := by exact_mod_cast hq |
| apply vague_open_null L.filter (fun i => nonaffinePart (S.multiLaw i)) L.multi L.multi_converges |
| (isOpen_lt continuous_const (hf.comp continuous_subtype_val)) |
| intro g hg hz |
| have he : ∀ᶠ i in atTop,C/(2:ℝ)^(criticalLowerExponent S.offset i)<q := |
| (tendsto_const_nhds.div_atTop S.power_lower_tendsto).eventually (gt_mem_nhds hqR) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [L.refines he] with i hi |
| symm |
| apply integral_nonaffine_scale_zero_of_bound _ _ _ _ _ (S.epsilon_pos i) g f q |
| · intro d hd; exact hz d (not_lt.mpr hd) |
| · intro j hj z |
| exact (hb _ _ z _ (by positivity)).trans ((div_le_div_of_nonneg_left hC (by positivity) |
| (by exact_mod_cast pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (Nat.le_add_right (criticalLowerExponent S.offset i) j))).trans hi.le) |
| lemma sample_constraint_aeterminal (f : DistanceArray → ℝ) (hf : Continuous f) (C : ℝ) (hC : 0 ≤ C) |
| (hb : ∀ k : ℝ,∀ hk : 0 ≤ k,∀ z : Torus,∀ n : ℕ,∀ hn : 0<n, |
| f (sampleArray k hk z n hn) ≤ C/(n:ℝ)) : ∀ᵐ d ∂L.terminal, f d.val ≤ 0 := by |
| apply ae_le_zero_of_positive_rational_null |
| intro q hq |
| have hqR : (0:ℝ)<q := by exact_mod_cast hq |
| apply vague_open_null L.filter (fun i => nonaffinePart (S.terminalLaw i)) L.terminal L.terminal_converges |
| (isOpen_lt continuous_const (hf.comp continuous_subtype_val)) |
| intro g hg hz |
| have he : ∀ᶠ i in atTop,C/(2:ℝ)^(S.exponent i)<q := |
| (tendsto_const_nhds.div_atTop S.power_terminal_tendsto).eventually (gt_mem_nhds hqR) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [L.refines he] with i hi |
| symm |
| apply integral_nonaffine_scale_zero_of_bound _ _ _ _ _ (S.epsilon_pos i) g f q |
| · intro d hd; exact hz d (not_lt.mpr hd) |
| · intro j hj z |
| exact (hb _ _ z _ (by positivity)).trans ((div_le_div_of_nonneg_left hC (by positivity) |
| (by exact_mod_cast pow_le_pow_right₀ (by norm_num : (1:ℝ) ≤ 2) (Nat.le_add_right (S.exponent i) j))).trans hi.le) |
| lemma unit_aemulti : ∀ᵐ d ∂L.multi, UnitArray d.val := by |
| have h (s t : DyadicTime) : ∀ᵐ d ∂L.multi,d.val.val s t-|(t:ℝ)-(s:ℝ)| ≤ 0 := by |
| apply S.sample_constraint_aemulti L (fun d => d.val s t-|(t:ℝ)-(s:ℝ)|) |
| ((continuous_arrayEval s t).sub continuous_const) 1 (by norm_num) |
| intro k hk z n hn |
| change rescaledDistance k z n s t-|(t:ℝ)-(s:ℝ)| ≤ 1/(n:ℝ) |
| linarith [rescaledDistance_bound k hk z n hn s t] |
| filter_upwards [ae_all_iff.mpr (fun s => ae_all_iff.mpr (h s))] with d hd |
| intro s t |
| linarith [hd s t] |
| lemma unit_aeterminal : ∀ᵐ d ∂L.terminal, UnitArray d.val := by |
| have h (s t : DyadicTime) : ∀ᵐ d ∂L.terminal,d.val.val s t-|(t:ℝ)-(s:ℝ)| ≤ 0 := by |
| apply S.sample_constraint_aeterminal L (fun d => d.val s t-|(t:ℝ)-(s:ℝ)|) |
| ((continuous_arrayEval s t).sub continuous_const) 1 (by norm_num) |
| intro k hk z n hn |
| change rescaledDistance k z n s t-|(t:ℝ)-(s:ℝ)| ≤ 1/(n:ℝ) |
| linarith [rescaledDistance_bound k hk z n hn s t] |
| filter_upwards [ae_all_iff.mpr (fun s => ae_all_iff.mpr (h s))] with d hd |
| intro s t |
| linarith [hd s t] |
| lemma tree_aemulti : ∀ᵐ d ∂L.multi, TreeArray d.val := by |
| have h (v x y z : DyadicTime) : ∀ᵐ d ∂L.multi, |
| min (d.val.val v x+d.val.val v y-d.val.val x y) (d.val.val v y+d.val.val v z-d.val.val y z)- |
| (d.val.val v x+d.val.val v z-d.val.val x z) ≤ 0 := by |
| have hc (a b c : DyadicTime) : Continuous (fun d : DistanceArray => d.val a b+d.val a c-d.val b c) := |
| ((continuous_arrayEval a b).add (continuous_arrayEval a c)).sub (continuous_arrayEval b c) |
| exact S.sample_constraint_aemulti L _ (((hc v x y).min (hc v y z)).sub (hc v x z)) 3 (by norm_num) |
| (fun k hk q n hn => rescaledDistance_tree k hk q n hn v x y z) |
| filter_upwards [ae_all_iff.mpr (fun v => ae_all_iff.mpr (fun x => ae_all_iff.mpr |
| (fun y => ae_all_iff.mpr (h v x y))))] with d hd |
| intro v x y z |
| linarith [hd v x y z] |
| lemma tree_aeterminal : ∀ᵐ d ∂L.terminal, TreeArray d.val := by |
| have h (v x y z : DyadicTime) : ∀ᵐ d ∂L.terminal, |
| min (d.val.val v x+d.val.val v y-d.val.val x y) (d.val.val v y+d.val.val v z-d.val.val y z)- |
| (d.val.val v x+d.val.val v z-d.val.val x z) ≤ 0 := by |
| have hc (a b c : DyadicTime) : Continuous (fun d : DistanceArray => d.val a b+d.val a c-d.val b c) := |
| ((continuous_arrayEval a b).add (continuous_arrayEval a c)).sub (continuous_arrayEval b c) |
| exact S.sample_constraint_aeterminal L _ (((hc v x y).min (hc v y z)).sub (hc v x z)) 3 (by norm_num) |
| (fun k hk q n hn => rescaledDistance_tree k hk q n hn v x y z) |
| filter_upwards [ae_all_iff.mpr (fun v => ae_all_iff.mpr (fun x => ae_all_iff.mpr |
| (fun y => ae_all_iff.mpr (h v x y))))] with d hd |
| intro v x y z |
| linarith [hd v x y z] |
| end CriticalScaleSequence |
| end StandardMapEntropy |
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| end OAI |
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