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92416a2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import OAI.Dynamics.ConditionalShuffle.HybridRaw
namespace OAI
noncomputable section
open scoped Classical
namespace Thorp.Conditional.Hybrid
open Revealed.Split Revealed.Disintegration
abbrev Public (d t : ℕ) := Fin t → Option (Coins d)
def energy (d : ℕ) (v : RawState (d+1)) (t : ℕ) (s : Fin t → Bool) : ℝ :=
mean (fun ω : History (d+1) t => rawEnergy (raw d v t s ω))
lemma tagged_deviation {ι : Type*} (d : ℕ)
(B : Position (d+1) → Bool) (tag : Position (d+1)) (ht : B tag = true)
(e : ι → Position (d+1)) (he : ∀ i, B (e i) = false) (t : ℕ) (s : Fin t → Bool)
(f : Public (d+1) t → (Fin (t+1) → ι → Position (d+1)) → Position (d+1) → ℝ)
(hf : ∀ q p y, |f q p y| ≤ 1) :
|mean (fun ω : History (d+1) t =>
f (publicHistory s ω) (exposedPath (d+1) e t ω) (run (d+1) t ω tag) -
(1 / (freeCount B : ℝ)) * ∑ y, if B ((run (d+1) t ω).symm y) then
f (publicHistory s ω) (exposedPath (d+1) e t ω) y else 0)| ≤
Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) *
energy d (initialState (d+1) B tag ht).raw t s) := by
have hh := flow_joint d (initialState (d+1) B tag ht).raw e he t s f
have hl : mean (fun ω : History (d+1) t =>
f (publicHistory s ω) (exposedPath (d+1) e t ω) (run (d+1) t ω tag) -
(1 / (freeCount B : ℝ)) * ∑ y, if B ((run (d+1) t ω).symm y) then
f (publicHistory s ω) (exposedPath (d+1) e t ω) y else 0) =
mean (fun ω : History (d+1) t => ∑ x, (initialState (d+1) B tag ht).raw.weight x *
f (publicHistory s ω) (exposedPath (d+1) e t ω) (run (d+1) t ω x)) := by
apply mean_congr; intro ω
change _ = ∑ x, centeredPoint B tag x *
f (publicHistory s ω) (exposedPath (d+1) e t ω) (run (d+1) t ω x)
rw [centeredPoint_test]
congr 2
simpa only [Equiv.symm_apply_apply] using
(Equiv.sum_comp (run (d+1) t ω) (fun y => if B ((run (d+1) t ω).symm y) then
f (publicHistory s ω) (exposedPath (d+1) e t ω) y else 0)).symm
rw [hl, hh]
exact mean_weighted_test_le _ _ (fun ω y => hf _ _ _)
lemma completion_error {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
(d : ℕ) (L : Sum ι α ≃ Position (d+1)) {k : ℕ} (e : Fin k → α)
(tag : α) (ht : freeOf e tag = true) (t : ℕ) (s : Fin t → Bool)
(f : Public (d+1) t → (Fin (t+1) → ι → Position (d+1)) → (Fin (k+1) → Position (d+1)) → ℝ)
(hf : ∀ q p z, |f q p z| ≤ 1) :
|mean (fun ω : History (d+1) t =>
f (publicHistory s ω) (splitPath L t ω) (splitTuple L t ω (Fin.snoc e tag))) -
mean (fun ω : History (d+1) t =>
pathComplete (f (publicHistory s ω)) (splitPath L t ω) (splitTuple L t ω e))| ≤
Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * energy d
(initialState (d+1) (avoid (Sum.elim (fun i => L (.inl i)) (fun j => L (.inr (e j)))))
(L (.inr tag)) (by rwa [avoid_equiv_right])).raw t s) := by
let z : Sum ι (Fin k) → Position (d+1) :=
Sum.elim (fun i => L (.inl i)) (fun j => L (.inr (e j)))
let B := avoid z
let F (q : Public (d+1) t) (p : Fin (t+1) → Sum ι (Fin k) → Position (d+1)) (y : Position (d+1)) :=
f q (fun a i => p a (.inl i)) (Fin.snoc (fun j => p (Fin.last t) (.inr j)) y)
have hh := tagged_deviation d B (L (.inr tag)) ((avoid_equiv_right L e tag).trans ht)
z (avoid_exposed z) t s F (fun q p y => hf _ _ _)
have ho (ω : History (d+1) t) :
(fun a i => exposedPath (d+1) z t ω a (.inl i)) = splitPath L t ω :=
exposedPath_comp (d+1) z Sum.inl t ω
have he (ω : History (d+1) t) :
(fun j => exposedPath (d+1) z t ω (Fin.last t) (.inr j)) = splitTuple L t ω e := by
rw [exposedPath_last]; rfl
have hF (q) (ω : History (d+1) t) (y : Position (d+1)) :
F q (exposedPath (d+1) z t ω) y = f q (splitPath L t ω) (Fin.snoc (splitTuple L t ω e) y) := by
change f _ _ _ = _; rw [ho, he]
have hB (ω : History (d+1) t) (y : Position (d+1)) :
B ((run (d+1) t ω).symm y) =
avoid (Sum.elim (splitPath L t ω (Fin.last t)) (splitTuple L t ω e)) y := by
rw [splitPath_last]
have hz : Sum.elim (fun i => run (d+1) t ω (L (.inl i))) (splitTuple L t ω e) =
(run (d+1) t ω) ∘ z := by funext i; cases i <;> rfl
rw [hz]; exact (avoid_perm z (run (d+1) t ω) y).symm
have hcount (ω : History (d+1) t) :
freeCount (avoid (Sum.elim (splitPath L t ω (Fin.last t)) (splitTuple L t ω e))) = freeCount B := by
rw [splitPath_last]
change freeCount (avoid (Sum.elim (fun i => (L.trans (run (d+1) t ω)) (.inl i))
(fun j => (L.trans (run (d+1) t ω)) (.inr (e j))))) = _
rw [freeCount_avoid_equiv, show freeCount B = freeCount (freeOf e) from freeCount_avoid_equiv L e]
simp_rw [hF, hB] at hh
rw [← mean_sub]
convert hh using 1
congr 1
apply mean_congr; intro ω
rw [splitTuple_snoc]
congr 1
exact congrArg (fun n : ℕ => (1/(n : ℝ)) * ∑ y,
if avoid (Sum.elim (splitPath L t ω (Fin.last t)) (splitTuple L t ω e)) y then
f (publicHistory s ω) (splitPath L t ω) (Fin.snoc (splitTuple L t ω e) y) else 0) (hcount ω)
theorem tuple_test {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α]
(d t m : ℕ) (s : Fin t → Bool) (ε : ℝ)
(hE : ∀ (B : Position (d+1) → Bool) (tag) (ht : B tag = true),
m ≤ freeCount B → freeCount B ≤ Fintype.card α →
energy d (initialState (d+1) B tag ht).raw t s ≤ ε)
(L : Sum ι α ≃ Position (d+1)) (k : ℕ) (hk : k+m ≤ Fintype.card α)
(e : Fin k → α) (he : Function.Injective e)
(f : Public (d+1) t → (Fin (t+1) → ι → Position (d+1)) → (Fin k → Position (d+1)) → ℝ)
(hf : ∀ q p z, |f q p z| ≤ 1) :
|mean (fun ω : History (d+1) t => f (publicHistory s ω) (splitPath L t ω) (splitTuple L t ω e)) -
mean (fun ω : History (d+1) t => mean (fun σ : Equiv.Perm α =>
f (publicHistory s ω) (splitPath L t ω) (splitTuple L t ω (σ ∘ e))))| ≤
(k : ℝ) * Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * ε) := by
induction k with
| zero =>
have hz (ω : History (d+1) t) (σ : Equiv.Perm α) :
splitTuple L t ω (σ ∘ e) = splitTuple L t ω e := Subsingleton.elim _ _
simp_rw [hz, mean_const]
simp
| succ k ih =>
have he0 : Function.Injective (Fin.init e) := by
intro i j hij; exact Fin.castSucc_inj.mp (he hij)
have ht : freeOf (Fin.init e) (e (Fin.last k)) = true := by
apply (freeOf_true _ _).mpr; intro i hi
exact Fin.castSucc_ne_last i (he hi)
let B := avoid (Sum.elim (fun i => L (.inl i)) (fun j => L (.inr (Fin.init e j))))
have htag : B (L (.inr (e (Fin.last k)))) = true := (avoid_equiv_right _ _ _).trans ht
have hcount : freeCount B = Fintype.card α - k := by
rw [show freeCount B = freeCount (freeOf (Fin.init e)) from freeCount_avoid_equiv L (Fin.init e),
freeCount_freeOf _ he0]
have henergy := hE B _ htag (by omega) (by omega)
have hz := completion_error d L (Fin.init e) (e (Fin.last k)) ht t s f hf
rw [Fin.snoc_init_self] at hz
have hestimate : Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) *
energy d (initialState (d+1) B (L (.inr (e (Fin.last k)))) htag).raw t s) ≤
Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * ε) :=
Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left henergy (Nat.cast_nonneg _))
have hnext := ih (by omega) (Fin.init e) he0
(fun q => pathComplete (f q)) (fun q => pathComplete_bound (f q) (hf q))
have hu (ω : History (d+1) t) := uniform_splitTuple_completion L e he t ω (f (publicHistory s ω))
simp_rw [hu]
have htri := abs_sub_le
(mean (fun ω : History (d+1) t => f (publicHistory s ω) (splitPath L t ω) (splitTuple L t ω e)))
(mean (fun ω : History (d+1) t => pathComplete (f (publicHistory s ω)) (splitPath L t ω)
(splitTuple L t ω (Fin.init e))))
(mean (fun ω : History (d+1) t => mean (fun σ : Equiv.Perm α =>
pathComplete (f (publicHistory s ω)) (splitPath L t ω) (splitTuple L t ω (σ ∘ Fin.init e)))))
have hz' := hz.trans hestimate
push_cast
nlinarith
lemma assignment_test {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [Nonempty α]
(d t m : ℕ) (s : Fin t → Bool) (ε : ℝ)
(hE : ∀ (B : Position (d+1) → Bool) (tag) (ht : B tag = true),
m ≤ freeCount B → freeCount B ≤ Fintype.card α →
energy d (initialState (d+1) B tag ht).raw t s ≤ ε)
(L : Sum ι α ≃ Position (d+1)) (k : ℕ) (hk : k+m ≤ Fintype.card α)
(e : Fin k → α) (he : Function.Injective e)
(f : Public (d+1) t → (Fin (t+1) → ι → Position (d+1)) → (Fin k → α) → ℝ)
(hf : ∀ q p z, |f q p z| ≤ 1) :
|mean (fun ω : History (d+1) t => f (publicHistory s ω) (splitPath L t ω)
(assignment (L.trans (run (d+1) t ω)) ∘ e)) -
mean (fun ω : History (d+1) t => mean (fun σ : Equiv.Perm α =>
f (publicHistory s ω) (splitPath L t ω) (σ ∘ e)))| ≤
(k : ℝ) * Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * ε) := by
let F (q : Public (d+1) t) (p : Fin (t+1) → ι → Position (d+1)) (z : Fin k → Position (d+1)) :=
f q p (fun j => indexRight (frameFor L (p (Fin.last t))) (z j))
have hh := tuple_test d t m s ε hE L k hk e he F (fun q p z => hf _ _ _)
have hF (ω : History (d+1) t) (a : Fin k → α) :
F (publicHistory s ω) (splitPath L t ω) (splitTuple L t ω a) =
f (publicHistory s ω) (splitPath L t ω) (assignment (L.trans (run (d+1) t ω)) ∘ a) := by
dsimp only [F, splitTuple]
rw [splitPath_last, indexRight_tuple]
simp_rw [hF] at hh
have hu (ω : History (d+1) t) := mean_perm_comp (assignment (L.trans (run (d+1) t ω)))
e (f (publicHistory s ω) (splitPath L t ω))
simp_rw [hu] at hh
exact hh
lemma marginal_bound {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] [Nonempty α]
(d t m : ℕ) (s : Fin t → Bool) (ε : ℝ)
(hE : ∀ (B : Position (d+1) → Bool) (tag) (ht : B tag = true),
m ≤ freeCount B → freeCount B ≤ Fintype.card α →
energy d (initialState (d+1) B tag ht).raw t s ≤ ε)
(L : Sum ι α ≃ Position (d+1)) (k : ℕ) (hk : k+m ≤ Fintype.card α)
(e : Fin k → α) (he : Function.Injective e) :
let μ := fairMass (fun ω : History (d+1) t =>
((publicHistory s ω, splitPath L t ω), assignment (L.trans (run (d+1) t ω))))
(∑ p, marginal μ p * tv (Trim.push (conditional μ p) (fun g : Equiv.Perm α => g ∘ e))
(fairMass (fun g : Equiv.Perm α => g ∘ e))) ≤
(k : ℝ) / 2 * Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * ε) := by
have hh := conditional_projection_le_of_test
(fun ω : History (d+1) t => (publicHistory s ω, splitPath L t ω))
(fun ω : History (d+1) t => assignment (L.trans (run (d+1) t ω)))
(fun g : Equiv.Perm α => g ∘ e) ((k : ℝ) * Real.sqrt ((Fintype.card (Position (d+1)) : ℝ) * ε))
(fun f hf => assignment_test d t m s ε hE L k hk e he
(fun q p z => f ((q,p),z)) (fun q p z => hf _))
exact hh.trans_eq (by ring)
end Thorp.Conditional.Hybrid
end
end OAI
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