| import OAI.Dynamics.ConditionalShuffle.ResetEnergy |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Classical |
| open Filter Topology |
| namespace Thorp.Conditional |
| theorem prefix_test_energy (d t m : ℕ) (ε : ℝ) |
| (hE : ∀ (B : Position (d+2) → Bool) (tag) (ht : B tag = true), |
| m ≤ freeCount B → expectedEnergy (d+1) (initialState (d+2) B tag ht) t ≤ ε) |
| (k : ℕ) (hk : k + m ≤ Fintype.card (Position (d + 2))) |
| (e : Fin k → Position (d + 2)) (he : Function.Injective e) |
| (f : (Fin k → Position (d + 2)) → ℝ) (hf : ∀ z, |f z| ≤ 1) : |
| |mean (fun ω : History (d + 2) t => f ((run (d + 2) t ω) ∘ e)) - |
| mean (fun g : State (d + 2) => f (g ∘ e))| ≤ |
| (k : ℝ) * Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) * |
| ε) := by |
| induction k with |
| | zero => |
| have hz (g : State (d + 2)) : g ∘ e = e := by funext i; exact Fin.elim0 i |
| simp only [hz, mean_const, sub_self, abs_zero, Nat.cast_zero, zero_mul, le_refl] |
| | 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) |
| have hcount : m ≤ freeCount (freeOf (Fin.init e)) := by |
| rw [freeCount_freeOf _ he0] |
| omega |
| have henergy := hE (freeOf (Fin.init e)) (e (Fin.last k)) ht hcount |
| have hz := prefix_one_step_bound (d + 1) t k e he f hf |
| have hestimate : |
| Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) * |
| expectedEnergy (d + 1) (initialState (d + 2) (freeOf (Fin.init e)) (e (Fin.last k)) ht) |
| t) ≤ |
| Real.sqrt ((Fintype.card (Position (d + 2)) : ℝ) * |
| ε) := by |
| apply Real.sqrt_le_sqrt |
| apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _) |
| exact henergy |
| have hnext := ih (by omega) (Fin.init e) he0 (completeTest f) (completeTest_bound f hf) |
| rw [uniform_prefix_step k e he f] |
| have htri := abs_sub_le |
| (mean (fun ω : History (d + 2) t => f ((run (d + 2) t ω) ∘ e))) |
| (mean (fun ω : History (d + 2) t => completeTest f ((run (d + 2) t ω) ∘ Fin.init e))) |
| (mean (fun g : State (d + 2) => completeTest f (g ∘ Fin.init e))) |
| have hz' := hz.trans hestimate |
| push_cast |
| nlinarith |
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| lemma reset_sqrt_bound (D : ℕ) : |
| Real.sqrt ((2^D : ℝ)*(1/2 : ℝ)^(20*D)) ≤ (1/2 : ℝ)^(9*D) := by |
| apply (Real.sqrt_le_left (by positivity)).mpr |
| have he : (2^D : ℝ)*(1/2 : ℝ)^(20*D) = (1/2 : ℝ)^(19*D) := by |
| rw [pow_mul, ← mul_pow, pow_mul]; norm_num |
| rw [he, ← pow_mul] |
| apply pow_le_pow_of_le_one (by norm_num) (by norm_num) |
| omega |
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| theorem reset_partial_mixing (ρ : ℝ) (hρ : 0 < ρ) (hρ₁ : ρ ≤ 1) : |
| ∀ᶠ d : ℕ in atTop, ∀ (b : ℕ) (e : Fin b → Position (d+2)), |
| Function.Injective e → (b : ℝ) ≤ (1-ρ)*(2^(d+2) : ℝ) → |
| tv (fairMass (fun ω : History (d+2) (2*resetS ρ*(d+2)) => |
| run (d+2) (2*resetS ρ*(d+2)) ω ∘ e)) |
| (fairMass (fun g : State (d+2) => g ∘ e)) ≤ (1/2 : ℝ)^(8*(d+2)) := by |
| filter_upwards [reset_energy_eventually ρ hρ hρ₁] with d hd |
| intro b e he hb |
| have hbN : b ≤ Fintype.card (Position (d+2)) := by |
| simpa only [Fintype.card_fin] using Fintype.card_le_of_injective e he |
| let m := Fintype.card (Position (d+2)) - b |
| have hm : ρ * Fintype.card (Position (d+2)) ≤ (m : ℝ) := by |
| dsimp [m]; rw [Nat.cast_sub hbN] |
| rw [card_position] at *; push_cast at *; nlinarith |
| have hE (B : Position (d+2) → Bool) (tag) (ht : B tag = true) (hB : m ≤ freeCount B) : |
| expectedEnergy (d+1) (initialState (d+2) B tag ht) (2*resetS ρ*(d+2)) ≤ |
| (1/2 : ℝ)^(20*(d+2)) := by |
| apply hd |
| exact hm.trans (by exact_mod_cast hB) |
| have hh := fairMass_tv_le_of_test |
| (fun ω : History (d+2) (2*resetS ρ*(d+2)) => run (d+2) (2*resetS ρ*(d+2)) ω ∘ e) |
| (fun g : State (d+2) => g ∘ e) |
| ((b : ℝ)*Real.sqrt ((Fintype.card (Position (d+2)) : ℝ)*(1/2 : ℝ)^(20*(d+2)))) |
| (prefix_test_energy d (2*resetS ρ*(d+2)) m _ hE b (by dsimp [m]; omega) e he) |
| have hbN' : (b : ℝ) ≤ (2^(d+2) : ℝ) := by simpa only [card_position, Nat.cast_pow, Nat.cast_ofNat] using |
| (show (b : ℝ) ≤ (Fintype.card (Position (d+2)) : ℝ) by exact_mod_cast hbN) |
| have hmul : (2^(d+2) : ℝ)*(1/2 : ℝ)^(9*(d+2)) = (1/2 : ℝ)^(8*(d+2)) := by |
| rw [pow_mul, ← mul_pow, pow_mul]; norm_num |
| apply hh.trans |
| rw [card_position, Nat.cast_pow, Nat.cast_ofNat] |
| have hsq := reset_sqrt_bound (d+2) |
| have hproduct := mul_le_mul hbN' hsq (Real.sqrt_nonneg _) (by positivity) |
| rw [hmul] at hproduct |
| linarith [pow_nonneg (by norm_num : (0 : ℝ) ≤ 1/2) (8*(d+2))] |
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| end Thorp.Conditional |
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| end |
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| end OAI |
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