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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 | import OAI.Dynamics.ConditionalShuffle.CycleIterate
namespace OAI
noncomputable section
open scoped Classical
namespace Thorp.Conditional.Hybrid
open OverlayEnergy Revealed.Instrument
lemma raw_append (d n m : ℕ) (v : RawState (d+1))
(s : Fin n → Bool) (r : Fin m → Bool) (a : History (d+1) n) (b : History (d+1) m) :
raw d v (n+m) (Fin.append s r) (Fin.append a b) = raw d (raw d v n s a) m r b := by
induction m with
| zero =>
have hr : Fin.append s r = s := by funext i; exact Fin.append_left s r i
have hb : Fin.append a b = a := by funext i; exact Fin.append_left a b i
rw [hr, hb]; rfl
| succ m ih =>
obtain ⟨⟨q,r⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => Bool)).surjective r
obtain ⟨⟨c,b⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (m+1) => Coins (d+1))).surjective b
erw [Fin.append_snoc, Fin.append_snoc, raw_snoc, raw_snoc, ih]
lemma raw_public (d : ℕ) (v : RawState (d+1)) (t : ℕ) (ω : History (d+1) t) :
raw d v t (fun _ => false) ω = rawPermute v (run (d+1) t ω) := by
induction t with
| zero => rfl
| succ t ih =>
change rawPermute (raw d v t (fun _ => false) (Fin.init ω)) (step (d+1) (ω (Fin.last t))) = _
rw [ih, run_succ]
rfl
lemma raw_private (d : ℕ) (v : RawState (d+1)) (t : ℕ) (ω : History (d+1) t) :
raw d v t (fun _ => true) ω = rawIterate d v t ω := by
induction t with
| zero => rfl
| succ t ih =>
change rawNext d (raw d v t (fun _ => true) (Fin.init ω)) (ω (Fin.last t)) = _
rw [ih]; rfl
def periodLength (g : ℕ) := (9*(g+3)+200*(g+3))+(g+3)
lemma periodLength_eq (g : ℕ) : periodLength g = 210*(g+3) := by unfold periodLength; omega
def cycleSchedule (g : ℕ) : Fin (periodLength g) → Bool :=
Fin.append (Fin.append (fun _ : Fin (9*(g+3)) => false)
(fun _ : Fin (200*(g+3)) => false)) (fun _ : Fin (g+3) => true)
def tripleHistory (Ω : Type) (a b c : ℕ) :
(((Fin a → Ω) × (Fin b → Ω)) × (Fin c → Ω)) ≃ (Fin ((a+b)+c) → Ω) :=
(Equiv.prodCongr (Fin.appendEquiv a b) (Equiv.refl _)).trans (Fin.appendEquiv (a+b) c)
lemma tripleHistory_apply (Ω : Type) (a b c : ℕ) (ω) :
tripleHistory Ω a b c ω = Fin.append (Fin.append ω.1.1 ω.1.2) ω.2 := rfl
def cycleHistory (g : ℕ) : CycleCoins g ≃ (Fin (periodLength g) → Bits g) :=
tripleHistory (Bits g) (9*(g+3)) (200*(g+3)) (g+3)
lemma cycleHistory_apply (g : ℕ) (ω : CycleCoins g) :
cycleHistory g ω = Fin.append (Fin.append ω.1.1 ω.1.2) ω.2 :=
tripleHistory_apply (Bits g) _ _ _ ω
lemma raw_cycle (g : ℕ) (v : RawState (g+3)) (ω : CycleCoins g) :
raw (g+2) v (periodLength g) (cycleSchedule g) (cycleHistory g ω) = cycle g v ω := by
rw [cycleHistory_apply]
unfold cycleSchedule
erw [raw_append, raw_append, raw_public, raw_public, raw_private]
rfl
def repeatedSchedule (g n : ℕ) : Fin (periodLength g*n) → Bool :=
chunks Bool (periodLength g) n (fun _ => cycleSchedule g)
def repeatedHistory (g n : ℕ) : (Fin n → CycleCoins g) ≃ (Fin (periodLength g*n) → Bits g) :=
(Equiv.piCongrRight (fun _ : Fin n => cycleHistory g)).trans (chunks (Bits g) (periodLength g) n)
lemma repeatedHistory_apply (g n : ℕ) (ω : Fin n → CycleCoins g) :
repeatedHistory g n ω = chunks (Bits g) (periodLength g) n (fun i => cycleHistory g (ω i)) := by
rfl
lemma raw_cycles (g : ℕ) (v : RawState (g+3)) (n : ℕ) (ω : Fin n → CycleCoins g) :
raw (g+2) v (periodLength g*n) (repeatedSchedule g n) (repeatedHistory g n ω) = cycles g v n ω := by
induction n with
| zero => rfl
| succ n ih =>
obtain ⟨⟨c,ω⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (n+1) => CycleCoins g)).surjective ω
have hs : (fun _ : Fin (n+1) => cycleSchedule g) =
Fin.snoc (fun _ : Fin n => cycleSchedule g) (cycleSchedule g) := by
funext i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
have hh : (fun i : Fin (n+1) => cycleHistory g ((Fin.snoc ω c : Fin (n+1) → CycleCoins g) i)) =
Fin.snoc (fun i : Fin n => cycleHistory g (ω i)) (cycleHistory g c) := by
funext i; refine Fin.lastCases ?_ (fun j => ?_) i <;> simp
rw [repeatedHistory_apply, repeatedSchedule]
rw [hs]
change raw (g+2) v _ _ (chunks (Bits g) _ _
(fun i => cycleHistory g ((Fin.snoc ω c : Fin (n+1) → CycleCoins g) i))) = _
rw [hh, chunks_snoc, chunks_snoc]
erw [raw_append]
change raw (g+2) (raw (g+2) v _ (repeatedSchedule g n) (repeatedHistory g n ω))
_ (cycleSchedule g) (cycleHistory g c) = _
rw [ih, raw_cycle]
simp only [cycles, Fin.snocEquiv, Equiv.coe_fn_mk, Fin.init_snoc, Fin.snoc_last]
lemma repeated_energy (g : ℕ) (v : RawState (g+3))
(hw : ∀ x, v.free x = false → v.weight x = 0) (hz : ∑ x, v.weight x = 0)
(hf : Fintype.card (Position (g+3)) ≤ 64 * freeCount v.free)
(hk : freeCount v.free ≤ 2^g)
(hs : 256*(g+4) ≤ Fintype.card (Position (g+3))) (n : ℕ) :
energy (g+2) v (periodLength g*n) (repeatedSchedule g n) ≤ rawEnergy v * OverlayEnergy.coefficient g^n := by
have hh := mean_equiv (repeatedHistory g n)
(fun ω : History (g+3) (periodLength g*n) =>
rawEnergy (raw (g+2) v (periodLength g*n) (repeatedSchedule g n) ω))
change mean (fun ω : Fin n → CycleCoins g =>
rawEnergy (raw (g+2) v _ (repeatedSchedule g n) (repeatedHistory g n ω))) = _ at hh
simp only [raw_cycles] at hh
change mean (fun ω : History (g+3) (periodLength g*n) => _) ≤ _
have hi := cycles_expected_energy g v hw hz hf hk hs n
exact (mean_fintype_irrel _ _ _).le.trans (hh.symm.le.trans hi)
lemma repeated_point_energy (g : ℕ) (B : Position (g+3) → Bool) (tag) (ht : B tag = true)
(hf : Fintype.card (Position (g+3)) ≤ 64*freeCount B) (hk : freeCount B ≤ 2^g)
(hs : 256*(g+4) ≤ Fintype.card (Position (g+3))) (n : ℕ) :
energy (g+2) (initialState (g+3) B tag ht).raw (periodLength g*n) (repeatedSchedule g n) ≤
OverlayEnergy.coefficient g^n := by
have hh := repeated_energy g (initialState (g+3) B tag ht).raw
(initialState (g+3) B tag ht).supported (initialState (g+3) B tag ht).mass_zero hf hk hs n
exact hh.trans (mul_le_of_le_one_left (pow_nonneg (coefficient_nonneg g) n)
(initialState (g+3) B tag ht).energy_le_one)
end Thorp.Conditional.Hybrid
end
end OAI
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