| import OAI.Dynamics.StandardMap.BridgeDecay |
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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 |
|
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| structure BridgeGrid (Rw Rh : ℕ) where |
| p : ℕ |
| u : DyadicTime |
| v : DyadicTime |
| b : ℤ |
| N : ℕ |
| r : ℕ |
| large : 100≤ N |
| narrow : (r:ℝ)≤ (N:ℝ)/1000 |
| hu : ((2^p:ℕ):ℝ)*(u:ℝ)=(b:ℝ) |
| hv : ((2^p:ℕ):ℝ)*(v:ℝ)=(b:ℝ)+(N:ℝ) |
| sw : Fin Rw → DyadicTime |
| tw : Fin Rw → DyadicTime |
| sh : Fin Rh → DyadicTime |
| th : Fin Rh → DyadicTime |
| aw : Fin Rw → ℤ |
| nw : Fin Rw → ℕ |
| ah : Fin Rh → ℤ |
| nh : Fin Rh → ℕ |
| pw : ∀j,0< nw j |
| ph : ∀j,0< nh j |
| hwleft : ∀j,-(r:ℤ)≤ aw j |
| hwright : ∀j,aw j+(nw j:ℤ)≤ r |
| hhleft : ∀j,-(r:ℤ)≤ ah j-(N:ℤ) |
| hhright : ∀j,ah j+(nh j:ℤ)-(N:ℤ)≤ r |
| hsw : ∀j,((2^p:ℕ):ℝ)*(sw j:ℝ)=((aw j+b):ℤ) |
| htw : ∀j,((2^p:ℕ):ℝ)*(tw j:ℝ)=((aw j+b):ℤ)+(nw j:ℝ) |
| hsh : ∀j,((2^p:ℕ):ℝ)*(sh j:ℝ)=((ah j+b):ℤ) |
| hth : ∀j,((2^p:ℕ):ℝ)*(th j:ℝ)=((ah j+b):ℤ)+(nh j:ℝ) |
| lemma aligned_scale (p l : ℕ) (s : DyadicTime) (a : ℤ) |
| (hs : ((2^p:ℕ):ℝ)*(s:ℝ)=(a:ℝ)) : |
| ((2^(p+l):ℕ):ℝ)*(s:ℝ)=((a*2^l:ℤ):ℝ) := by |
| push_cast at hs ⊢ |
| rw [pow_add] |
| nlinarith [congrArg (fun x : ℝ => x*(2:ℝ)^l) hs] |
| lemma aligned_scale_end (p l : ℕ) (s : DyadicTime) (a : ℤ) (m : ℕ) |
| (hs : ((2^p:ℕ):ℝ)*(s:ℝ)=(a:ℝ)+(m:ℝ)) : |
| ((2^(p+l):ℕ):ℝ)*(s:ℝ)=((a*2^l:ℤ):ℝ)+((m*2^l:ℕ):ℝ) := by |
| have hh := aligned_scale p l s (a+(m:ℤ)) (by simpa only [Int.cast_add,Int.cast_natCast] using hs) |
| push_cast at hh ⊢ |
| nlinarith |
| lemma observation_nonneg_aligned {R : ℕ} (k : ℝ) (hk : 0≤ k) (n : ℕ) (hn : 0< n) |
| (s t : Fin R → DyadicTime) (a : Fin R → ℤ) (m : Fin R → ℕ) (hm : ∀i,0< m i) |
| (hs : ∀i,(n:ℝ)*(s i:ℝ)=(a i:ℝ)) (ht : ∀i,(n:ℝ)*(t i:ℝ)=(a i:ℝ)+(m i:ℝ)) |
| (F : (Fin R → ℝ) → ℝ) (hF0 : ∀d,0≤ arrayObservation s t F d) : |
| ∀z,0≤ shortfallObservation k a m F z := by |
| intro z |
| rw [←arrayObservation_sample k hk n hn s t a m hm hs ht F] |
| exact hF0 _ |
| lemma length_from_window (a : ℤ) (m r : ℕ) (h1 : -(r:ℤ)≤ a) (h2 : a+(m:ℤ)≤ r) : m≤ 2*r := by omega |
| lemma BridgeGrid.sample_bound {Rw Rh : ℕ} (B : BridgeGrid Rw Rh) |
| (k : ℝ) (hc : BridgeScalarControl k) (l : ℕ) |
| (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) |
| (η : ℝ) (hη : 0<η) (hδ : growthBase k^(-(1/10:ℝ)*((B.N*2^l: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) : |
| (∫ d,F d ∂sampleLaw k hc.nonneg (2^(B.p+l)) (by positivity))≤ D*bridgeRealConstant* |
| ((∫ d,arrayObservation B.sw B.tw Fw d ∂sampleLaw k hc.nonneg (2^(B.p+l)) (by positivity))+ |
| growthBase k^(-(1/10:ℝ)*((B.N*2^l:ℕ):ℝ)))* |
| ((∫ d,arrayObservation B.sh B.th Fh d ∂sampleLaw k hc.nonneg (2^(B.p+l)) (by positivity))/(η/2)) := by |
| have hp : (0:ℤ)≤ 2^l := by positivity |
| have h1 : 1≤ 2^l := Nat.one_le_pow l 2 (by omega) |
| have hwL (j) : -((B.r*2^l:ℕ):ℤ)≤ B.aw j*2^l := by |
| have hh := mul_le_mul_of_nonneg_right (B.hwleft j) hp |
| push_cast; nlinarith |
| have hwR (j) : B.aw j*2^l+((B.nw j*2^l:ℕ):ℤ)≤ ((B.r*2^l:ℕ):ℤ) := by |
| have hh := mul_le_mul_of_nonneg_right (B.hwright j) hp |
| push_cast; nlinarith |
| have hhL (j) : -((B.r*2^l:ℕ):ℤ)≤ B.ah j*2^l-((B.N*2^l:ℕ):ℤ) := by |
| have hh := mul_le_mul_of_nonneg_right (B.hhleft j) hp |
| push_cast; nlinarith |
| have hhR (j) : B.ah j*2^l+((B.nh j*2^l:ℕ):ℤ)-((B.N*2^l:ℕ):ℤ)≤ ((B.r*2^l:ℕ):ℤ) := by |
| have hh := mul_le_mul_of_nonneg_right (B.hhright j) hp |
| push_cast; nlinarith |
| have hsw (j) : ((2^(B.p+l):ℕ):ℝ)*(B.sw j:ℝ)=(((B.aw j*2^l+B.b*2^l):ℤ):ℝ) := by |
| simpa only [add_mul] using aligned_scale B.p l (B.sw j) (B.aw j+B.b) (B.hsw j) |
| have htw (j) : ((2^(B.p+l):ℕ):ℝ)*(B.tw j:ℝ)=(((B.aw j*2^l+B.b*2^l):ℤ):ℝ)+((B.nw j*2^l:ℕ):ℝ) := by |
| simpa only [add_mul] using aligned_scale_end B.p l (B.tw j) (B.aw j+B.b) (B.nw j) (B.htw j) |
| have hsh (j) : ((2^(B.p+l):ℕ):ℝ)*(B.sh j:ℝ)=(((B.ah j*2^l+B.b*2^l):ℤ):ℝ) := by |
| simpa only [add_mul] using aligned_scale B.p l (B.sh j) (B.ah j+B.b) (B.hsh j) |
| have hth (j) : ((2^(B.p+l):ℕ):ℝ)*(B.th j:ℝ)=(((B.ah j*2^l+B.b*2^l):ℤ):ℝ)+((B.nh j*2^l:ℕ):ℝ) := by |
| simpa only [add_mul] using aligned_scale_end B.p l (B.th j) (B.ah j+B.b) (B.nh j) (B.hth j) |
| have hnonnegW : ∀z,0≤ shortfallObservation k (fun j => B.aw j*2^l) (fun j => B.nw j*2^l) Fw z := by |
| intro z |
| have hh := observation_nonneg_aligned k hc.nonneg (2^(B.p+l)) (by positivity) B.sw B.tw |
| (fun j => B.aw j*2^l+B.b*2^l) (fun j => B.nw j*2^l) (fun j => Nat.mul_pos (B.pw j) (by positivity)) hsw htw Fw hW0 |
| (torusIter k (-(B.b*2^l)) z) |
| rw [←shortfallObservation_shift,←torusIter_add,add_neg_cancel,torusIter_zero] at hh |
| exact hh |
| have hnonnegH : ∀z,0≤ shortfallObservation k (fun j => B.ah j*2^l) (fun j => B.nh j*2^l) Fh z := by |
| intro z |
| have hh := observation_nonneg_aligned k hc.nonneg (2^(B.p+l)) (by positivity) B.sh B.th |
| (fun j => B.ah j*2^l+B.b*2^l) (fun j => B.nh j*2^l) (fun j => Nat.mul_pos (B.ph j) (by positivity)) hsh hth Fh hH0 |
| (torusIter k (-(B.b*2^l)) z) |
| rw [←shortfallObservation_shift,←torusIter_add,add_neg_cancel,torusIter_zero] at hh |
| exact hh |
| apply sample_bridge_integral k hc (2^(B.p+l)) (B.N*2^l) (B.r*2^l) (by positivity) |
| (le_trans B.large (Nat.le_mul_of_pos_right _ (by positivity))) |
| (by push_cast; nlinarith [mul_le_mul_of_nonneg_right B.narrow (show (0:ℝ)≤ 2^l by positivity)]) |
| B.u B.v (B.b*2^l) (aligned_scale B.p l B.u B.b B.hu) (aligned_scale_end B.p l B.v B.b B.N B.hv) |
| B.sw B.tw B.sh B.th (fun j => B.aw j*2^l) (fun j => B.nw j*2^l) Fw |
| (fun j => B.ah j*2^l) (fun j => B.nh j*2^l) Fh |
| (fun j => Nat.mul_pos (B.pw j) (by positivity)) (fun j => Nat.mul_pos (B.ph j) (by positivity)) |
| hsw htw hsh hth L hFw hFh |
| (fun j => length_from_window _ _ _ (hwL j) (hwR j)) |
| (fun j => length_from_window (B.ah j*2^l-((B.N*2^l:ℕ):ℤ)) _ _ (hhL j) (by linarith [hhR j])) |
| (fun j x hx hx' => by simpa only [sub_zero] using window_indices (a := B.aw j*2^l) (b := 0) (r := B.r*2^l) (by simpa only [sub_zero] using hwL j) (by simpa only [sub_zero] using hwR j) x hx hx') |
| (fun j x hx hx' => window_indices (hhL j) (hhR j) x hx hx') |
| hnonnegW hnonnegH hlog hLM η hη hδ F hF D hD hFW hzero |
| end StandardMapEntropy |
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| end OAI |
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