| import OAI.Analysis.FourierLLogL.Procedure.Deficit |
| import OAI.Analysis.FourierLLogL.Thermal.Posterior |
| /-! # Cost of revealing the new main observation -/ |
|
|
| open Function Set Filter |
| open scoped Topology BigOperators ComplexConjugate |
| noncomputable section |
| namespace OAI |
|
|
| namespace FourierLLogL.Dyadic.Procedure |
| open Law Node Real Period Compress.Params Thermal Fund Scene Fork |
| variable {M J j : ℕ} {s : Scene M} |
| open scoped Classical |
|
|
| private def measureRate (p : Node) (A : Form M) (z v : Grid M) := |
| side M * (boltz A (z-v)* boltz (place p M) (start p M-v)) |
|
|
| private lemma weight_pos' (p : Node) (A : Form M) (z v : Grid M) : |
| 0 < measureRate p A z v := |
| mul_pos (side_pos _) (mul_pos (boltz_pos _ _) (boltz_pos _ _)) |
|
|
| private lemma rate_sum_pos (p : Node) (A : Form M) (z : Grid M) : |
| 0 < ∑ v, measureRate p A z v := |
| Finset.sum_pos (fun _ _ => weight_pos' _ _ _ _) Finset.univ_nonempty |
|
|
| private lemma rate_sum_le (p : Node) (A : Form M) (hA : 0 ≤ A) (z : Grid M) : |
| ∑ v, measureRate p A z v ≤ exp (-F (A+place p M)) := by |
| let s : Profile M := [(A,z), (place p M,start p M)] |
| have ht : mixForm s=A+place p M := by simp [s,mixForm] |
| have hV (x : Grid M) : measureRate p A z x = side M*tune s x := by |
| unfold s measureRate; rw [tune_cons,tune_cons] |
| simp [tune,sumE] |
| have H := profile_envelope (s:=s) (by |
| intro i hi |
| simp only [s,List.mem_cons,List.not_mem_nil,or_false] at hi |
| rcases hi with h|h <;> (subst i; first | exact hA | exact place_valid _ _)) 0 |
| simp_rw [hV,← Finset.mul_sum] |
| unfold fc at H |
| simp only [ht,neg_zero,chi_zero_left,mul_one,expU_real,Complex.norm_real,Real.norm_eq_abs] at H |
| change expU (fun x => tune s x) ≤ _ |
| rw [F_origin]; exact (le_abs_self _).trans H |
|
|
| private def matchLabel (p : Node) (A : Form M) (z : Grid M) : Law (Grid M) := |
| proportional _ (fun i => (weight_pos' p A z i).le) (rate_sum_pos p A z) |
|
|
| private lemma match_pos (p : Node) (A : Form M) (z v : Grid M) : |
| 0 < matchLabel p A z v := |
| div_pos (weight_pos' _ _ _ _) (rate_sum_pos _ _ _) |
|
|
| private lemma KL_bound (p : Node) (A B : Form M) (hA : 0 ≤ A) (h : A+place p M ≤ B) |
| (x z : Grid M) : |
| let q := matchLabel p A |
| (channel B x).kl (q z) ≤ F B-F (A+place p M) + |
| (channel B x).Ex (fun y => form A (z-y) + form (place p M) (start p M-y)) := by |
| intro q |
| have hB : 0 ≤ B := (add_nonneg hA (place_valid _ _)).trans h |
| rw [kl_ex] |
| conv_rhs => lhs; rw [← Ex_const (channel B x) (F B-F (A+place p M))] |
| rw [← Ex_add]; apply Ex_mono; intro i |
| let S := place p M; let t := start p M |
| change log (channel B x i)-log (measureRate p A z i/∑ v, measureRate p A z v) ≤ |
| F B-F (A+S) + (form A (z-i)+form S (t-i)) |
| have H := log_le_log (rate_sum_pos p A z) (rate_sum_le p A hA z) |
| rw [log_exp] at H |
| rw [ch_K,K_eq,log_mul (side_pos _).ne' (exp_pos _).ne',log_exp, |
| Real.log_div (weight_pos' _ _ _ _).ne' (rate_sum_pos _ _ _).ne', |
| measureRate, log_mul (side_pos _).ne' (mul_pos (boltz_pos _ _) (boltz_pos _ _)).ne', |
| log_mul (boltz_pos _ _).ne' (boltz_pos _ _).ne',boltz,boltz,log_exp,log_exp] |
| change _ ≤ -F (A+S) at H |
| unfold S t at * |
| linarith [form_nonneg hB (i-x)] |
|
|
| theorem added_deficit (s : Scene M) (p : Node) (A B : Form M) |
| (hp : s.here.ok M) (he : s.here=p) (hh : 0 ≤ s.wage) (hm : 0 < s.mass) |
| (hA : 0 ≤ A) (hB : A+place p M ≤ B) : |
| deficit s (channel B) ≤ deficit s (channel A)+ |
| (F B-F (A+place p M)+4*(label A).Ex (form A)+264) := by |
| subst p |
| let q := s.here; let S := place q M |
| let k := channel A; let v := channel B |
| let T := matchLabel q A |
| let P := s.actor; let H := P.withNext k |
| let x := F B-F (A+S) |
| let l := (label A).Ex (form A) |
| let y := x+4*l+264 |
| have hi : H.Ex (fun z => (v z.1).kl (T z.2)) ≤ y := by |
| unfold H; rw [Ex_withNext] |
| have hhA : 0 ≤ S := place_valid q _ |
| have hx : S ≤ B := le_trans (le_add_of_nonneg_left hA) hB |
| have hv : A ≤ B := le_trans (le_add_of_nonneg_right hhA) hB |
| let m := q.start M |
| have hm' (i) (hi : 0 < P i) : |
| (k i).Ex (fun z => (v i).kl (T z)) ≤ y := by |
| have he : i∈s.here.block M := piece_support hp (actor_def hp hh hm) _ hi |
| have hE := place_internal (start_mem hp.1 hp.2) he |
| have hh (z w : Grid M) : form A (z-w)+form S (m-w) ≤ |
| 2*(form A (z-i)+form S (m-i))+ 2*(form A (i-w)+form S (i-w)) := by |
| have h1 := form_triangle hA (z-i) (i-w) |
| have h2 := form_triangle hhA (m-i) (i-w) |
| rw [sub_add_sub_cancel] at h1 h2 |
| linarith |
| have hG (v : Form M) (A : Form M) : |
| (channel A i).Ex (fun y => form v (i-y)) = (label A).Ex (form v) := by |
| have h1 (y : Grid M) : form v (i-y) = form v (y-i) := by |
| rw [show i-y= -(y-i) by ring, form_even _ _] |
| simp_rw [h1]; exact ch_noise _ _ _ |
| have Hta (z : Grid M) : |
| (v i).kl (T z) ≤ x+2*(form A (z-i)+form S (m-i))+ |
| 2*(l+100) := by |
| have hi := KL_bound q A B hA hB i z |
| apply hi.trans |
| have he := (v i).Ex_mono (hh z) |
| rw [Ex_add _ (f:=fun _ => 2*(form A (z-i)+form S (m-i))), Ex_const,Ex_lmul,Ex_add (f:=fun w => form A (i-w))] at he |
| unfold v at he |
| rw [hG,hG] at he |
| have h1 := S_energy q.invlen q.invlen_ge (loc_scale hp.2) B hx |
| have h2 := form_energy_mono hA hv hA |
| unfold x l m S q place at * |
| linarith |
| apply le_trans ((k i).Ex_mono Hta); unfold k |
| rw [Ex_add,Ex_add,Ex_lmul,Ex_add,Ex_const,Ex_const,Ex_const,ch_noise A (form A)] |
| unfold y x S m l q at *; linarith |
| exact (P.Ex_mono_mem hm').trans (P.Ex_const y).le |
| have he := (replace_signal P k v T (match_pos q A)).trans hi |
| unfold deficit; change levels s-P.privateH v ≤ levels s-P.privateH k+y |
| linarith |
|
|
| lemma priced_change : |
| ∀ᶠ j : ℕ in atTop, ∀ M J : ℕ, ∀ s : Scene M, ∀ f : Grid M → ℝ, |
| ∀ u : Update j M s.here, Safe s j J f → Safe (s.win u) j J f → 0 < (s.win u).mass → |
| u.Good capA s.fund.mood s.actor → |
| let A := s.fund.mood |
| deficit (s.win u) (channel (u.next A)) ≤ deficit (s.win u) (channel A) + 500*u.credit := by |
| filter_upwards [h_large 10,stiff_budget capA cap_ok] with j hp hx M J s f u hs hS hm hU |
| intro A |
| have H := added_deficit (win s u) s.here A (u.next A) hS.within rfl hS.positive hm |
| hs.capital.effects.nonneg (le_add_of_nonneg_right hU.positive) |
| have hh := hs.capital.effects.energy |
| let a := Thermal.H j |
| let d := height j |
| have he : 1 ≤ a := H_ge _ |
| have ht : 0 ≤ plateErr j := by unfold plateErr; exact div_nonneg (d_pos _).le (pow_nonneg (H_pos _).le _) |
| have hy := hU.reserves.1 |
| have hX := hs.capital.under |
| apply H.trans |
| have H : 4*(label A).Ex (form A)+264 ≤ 100*u.credit := by |
| have hw : |
| (label A).Ex (form A) ≤ 2*(capA*a*(plateErr j + 1000*a)) := by |
| apply le_trans hh |
| calc |
| _ ≤ (capA*a)*(2*(plateErr j+1000*a)) := by gcongr <;> linarith |
| _ = _ := by ring |
| change capA*a*(plateErr j + 1000*a)+2000 ≤ d/a^4 at hx |
| have hZ : d/a^4 ≤ u.credit := by |
| apply le_trans _ hy; unfold d a |
| gcongr <;> first | exact (d_pos j).le | exact H_ge j | omega |
| have h0 : 0 ≤ capA*a*(plateErr j + 1000*a) := |
| mul_nonneg (mul_nonneg (le_trans zero_le_one cap_ok) (by linarith)) (by linarith) |
| linarith |
| linarith [hU.cost] |
|
|
| end FourierLLogL.Dyadic.Procedure |
|
|
| end OAI |
|
|
