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d95525c | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | import OAI.Analysis.FourierLLogL.Compress.Rent
/-! # Data processing for contextual cost -/
open Set Function Filter
open scoped BigOperators
noncomputable section
namespace OAI
namespace FourierLLogL.Compress
open Law Real
variable {A B V W G E Ω Ω' : Type} [Fintype A] [Fintype B]
[Fintype V] [Fintype W] [Fintype G] [Fintype E] [Nonempty E]
[Fintype Ω] [Fintype Ω'] [AddCommGroup G] {n : ℕ}
set_option linter.unusedSectionVars false
lemma Hcond_congr {P : Law Ω} {X Y : Ω → A} {W W' : Ω → B}
(h : ∀ a, 0 < P a → X a = Y a) (h' : ∀ a, 0 < P a → W a = W' a) :
P.Hcond X W = P.Hcond Y W' := by
unfold Hcond
rw [H_congr_mem P h']
congr 1
refine H_congr_mem P fun a ha => ?_
rw [h a ha,h' a ha]
lemma rent_proj_eq (tag : B → G) (w : B → V) (u : E → G) (c : E → ℝ)
(P : Law Ω) (X : Ω → Letters n A) (U : Ω → E) (σ : A → B) :
let Y := ctxt σ X
rent (tag ∘ σ) (w ∘ σ) u c P X U =
rent tag w u c P Y U +
((∑ i, P.Hcond (readP X i) (readP Y i)) -
P.Hcond X (fun x => (Y x, ans tag Y (fun a => u (U a)) x))) := by
intro Y
let Z := ans tag Y (fun a => u (U a))
have h1 (i : Fin n) :
P.Hcond (readP X i) ((w ∘ σ) ∘ readP X i) =
P.Hcond (readP X i) (readP Y i) + P.Hcond (readP Y i) (w ∘ readP Y i) := by
let R := (w ∘ readP Y i)
have he := Hcond_reduce P (readP X i) R σ
change P.Hcond _ R = _ + P.Hcond (readP Y i) R
rw [he]; congr 1
apply Hcond_right_partition
simp only [R,Y,ctxt,readP,comp_apply,Prod.mk.injEq]
grind
have h2 :
P.Hcond X (fun x => (Z x,ctxt w Y x)) =
P.Hcond X (fun x => (Y x,Z x)) +
P.Hcond Y (fun x => (Z x,ctxt w Y x)) := by
nth_rw 1 [Hcond_reduce P X _ (fun v i => σ (v i))]
congr 1
apply Hcond_right_partition
intro a b
simp only [Prod.mk.injEq,Y,ctxt,funext_iff]
grind
unfold rent
change _ - P.Hcond X (fun x => (Z x,ctxt w Y x))+_ = _
rw [h2]
simp_rw [h1,Finset.sum_add_distrib]
dsimp only [Z]
ring
lemma rent_projection (tag : B → G) (w : B → V) (u : E → G) (c : E → ℝ)
(P : Law Ω) (X : Ω → Letters n A) (U : Ω → E) (σ : A → B) :
rent tag w u c P (ctxt σ X) U ≤ rent (tag ∘ σ) (w ∘ σ) u c P X U := by
rw [rent_proj_eq]
refine le_add_of_nonneg_right (sub_nonneg.mpr ?_)
let Y := ctxt σ X
let Z (x : Ω) := (Y x,ans tag Y (fun x => u (U x)) x)
refine (Hcond_vec_le P X Z).trans (Finset.sum_le_sum fun i _ => ?_)
apply Hcond_antitone
intro a b h
exact congrFun (congrArg Prod.fst h) i
lemma rent_congr_mem (tag : A → G) (w : A → V) (u : E → G) (c : E → ℝ)
(P : Law Ω) {X Y : Ω → Letters n A} (U : Ω → E)
(h : ∀ x, 0 < P x → X x=Y x) :
rent tag w u c P X U = rent tag w u c P Y U := by
unfold rent
suffices H : ∑ i, P.Hcond (readP X i) (w ∘ readP X i) =
∑ i, P.Hcond (readP Y i) (w ∘ readP Y i) by
have hh (x : Ω) (hx : 0 < P x) :
(ans tag X (fun x => u (U x)) x, ctxt w X x) =
(ans tag Y (fun x => u (U x)) x, ctxt w Y x) := by
unfold ans ctxt; rw [h x hx]
rw [H]
have ht := Hcond_congr h (W:=fun x => (ans tag X (fun x => u (U x)) x, ctxt w X x))
(W':=fun x => (ans tag Y (fun x => u (U x)) x, ctxt w Y x)) hh
rw [ht]
refine Finset.sum_congr rfl fun i _ => ?_
apply Hcond_congr
· exact fun x hx => congrFun (h x hx) _
· exact fun x hx => congrArg w (congrFun (h x hx) _)
lemma opt_proj (tag : B → G) (w : B → V) (u : E → G) (c : E → ℝ)
(p : Law A) (σ : A → B) (n : ℕ) :
opt n (p.send σ) tag w u c = opt n p (tag ∘ σ) (w ∘ σ) u c := by
classical
apply le_antisymm
· obtain ⟨P,hP,he⟩ := (opt_spec n p (tag ∘ σ) (w ∘ σ) u c).1
rw [← he]
let X : WordSample n A E → Letters n A := Prod.fst
exact (opt_le_rent tag w u c (p.send σ) P (ctxt σ X) Prod.snd fun i => by
rw [← hP i,send_send]; rfl).trans (rent_projection tag w u c P _ _ _)
obtain ⟨P,hP,he⟩ := (opt_spec n (p.send σ) tag w u c).1
let X : WordSample n B E → Letters n B := Prod.fst
let Q x := pack (fun i : Fin n => p.after σ (X x i))
let R := P.withNext Q
let Y : WordSample n B E × (Letters n A) → Letters n A := Prod.snd
let B0 (z : WordSample n B E × Letters n A) := X z.1
let error (z : WordSample n B E × Letters n A) := z.1.2
have hr : hasRows p R Y := by
intro i
apply ext_of_Ex; intro f
rw [Ex_send]
change R.Ex (fun z => f (z.2 i)) = p.Ex f
unfold R
rw [Ex_withNext]
simp_rw [Q, Ex_pack_proj _ i f]
change P.Ex (fun x => (p.after σ (readP X i x)).Ex f) = _
rw [← Ex_send P (readP X i) (fun b => (p.after σ b).Ex f),show P.send (readP X i) = _ from hP i]
exact p.after_Ex_sum σ (fun _ => f)
have heq (x : WordSample n B E × Letters n A) (hx : 0 < R x) : ctxt σ Y x = B0 x := by
rcases x with ⟨a,b⟩
change 0 < P a * Q a b at hx
have hpos : 0 < P a ∧ 0 < Q a b :=
(mul_pos_iff.mp hx).resolve_right (by intro hh; linarith [hh.1,P.nonneg a])
funext i
have hi := (pack_pos (fun i => p.after σ (X a i)) b).mp hpos.2 i
have hmarg := P.send_pos (readP X i) hpos.1
rw [show P.send (readP X i) = _ from hP i] at hmarg
exact ((after_mem hmarg).mp hi).1
have hsrc : rent tag w u c R B0 error = opt n (p.send σ) tag w u c := by
have h1 := send_fst_withNext P Q
have hi := rent_send tag w u c R X Prod.snd Prod.fst
change rent tag w u c (R.send Prod.fst) X Prod.snd = _ at hi
rw [h1] at hi
exact hi.symm.trans he
refine (opt_le_rent (tag ∘ σ) (w ∘ σ) u c p R Y error hr).trans ?_
rw [rent_proj_eq]
let B1 := ctxt σ Y
let Z (b : WordSample n B E × Letters n A) :=
(B1 b, ans tag B1 (fun x => u (error x)) b)
let Z0 (b : WordSample n B E × Letters n A) :=
(B0 b, ans tag B0 (fun x => u (error x)) b)
change rent tag w u c R B1 error + (_ - R.Hcond Y Z) ≤ _
rw [rent_congr_mem tag w u c R error heq,hsrc]
suffices H : ∑ i, R.Hcond (readP Y i) (readP B1 i) ≤ R.Hcond Y Z by linarith
have hZ : R.Hcond Y Z = R.Hcond Y Z0 :=
Hcond_congr (fun _ _ => rfl) fun x hx => by unfold Z Z0 ans; rw [show B1 x = B0 x from heq x hx]
rw [hZ]
apply le_trans (b:=R.Hcond Y Prod.fst)
· have H1 : R.Hcond Y Prod.fst = P.Ex (fun x => (Q x).entropy) := by
have H := Hcond_withNext' P Q (fun _ y => y)
simpa only [H_self] using H
have H2 (i : Fin n) : R.Hcond (readP Y i) (readP B1 i) = p.Hcond (fun a => a) σ := by
rw [← hr i]
rw [Hcond_send]; rfl
have H3 (i : Fin n) :
P.Ex (fun x => (p.after σ (X x i)).entropy) = p.Hcond (fun a => a) σ := by
rw [Hcond_after]
simp_rw [H_self]
have hi := hP i
rw [← hi,Ex_send]; rfl
rw [H1]; simp_rw [Q,entropy_pack,H2,Ex_finset_sum,H3]; exact le_rfl
· apply Hcond_antitone
rintro ⟨a,b⟩ ⟨a',b'⟩ h
cases (show a=a' from h)
rfl
lemma opt_refined (tag : B → G) (w : B → V) (w' : A → W) (u : E → G) (c : E → ℝ)
(p : Law A) (σ : A → B) (n : ℕ) (g : W → V) (h : w ∘ σ = g ∘ w') :
opt n p (tag ∘ σ) w' u c ≤ opt n (p.send σ) tag w u c := by
rw [opt_proj tag w]
obtain ⟨P,hP,he⟩ := (opt_spec n p (tag ∘ σ) (w ∘ σ) u c).1
exact (opt_le_rent _ _ _ _ p P _ _ hP).trans
((rent_antitone _ w' (w ∘ σ) u c g h P Prod.fst Prod.snd).trans he.le)
end FourierLLogL.Compress
end OAI
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