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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.MarkedBits
namespace OAI
section
section
namespace CriticalPercolation.ReverseReference
open scoped Classical
lemma trajectory_dominance (a r v : ℕ → ℝ) (n : ℕ)
(hr : ∀ i < n, 0 ≤ r i) (hv : ∀ i < n, 0 ≤ v i)
(hstep : ∀ i < n, a i * r i ≤ a (i+1) * v i) :
a 0 * ∏ i : Fin n, r i.val ≤ a n * ∏ i : Fin n, v i.val := by
induction n with
| zero => simp
| succ n ih =>
have hh := ih (fun i hi => hr i (hi.trans (Nat.lt_succ_self n)))
(fun i hi => hv i (hi.trans (Nat.lt_succ_self n)))
(fun i hi => hstep i (hi.trans (Nat.lt_succ_self n)))
rw [Fin.prod_univ_castSucc,Fin.prod_univ_castSucc]
simp only [Fin.val_castSucc,Fin.val_last]
calc
_ = (a 0 * ∏ i : Fin n, r i.val) * r n := by ring
_ ≤ (a n * ∏ i : Fin n, v i.val) * r n := mul_le_mul_of_nonneg_right hh (hr n (Nat.lt_succ_self n))
_ = (a n*r n)*(∏ i : Fin n, v i.val) := by ring
_ ≤ (a (n+1)*v n)*(∏ i : Fin n, v i.val) := mul_le_mul_of_nonneg_right
(hstep n (Nat.lt_succ_self n)) (Finset.prod_nonneg fun i _ => hv i.val (i.isLt.trans (Nat.lt_succ_self n)))
_ = _ := by ring
lemma joint_trajectory_dominance {X M : Type*} [Fintype X] [Fintype M]
(μ : X → ℝ) (k : ℕ → X → M → ℝ) (u : ℕ → X → M → X)
(n : ℕ) (a : Fin n → X → M → ℝ) (σ : ℕ → X) (m : ℕ → M)
(m' : Fin n → M) (r v : Fin n → ℝ)
(hu : ∀ i < n, u i (σ (i+1)) (m i) = σ i)
(hr : ∀ i, 0 ≤ r i)
(hv : ∀ i, 0 ≤ k i.val (σ (i.val+1)) (m i.val) * a i (σ 0) (m' i) * v i)
(hs : ∀ i : Fin n, μ (σ i.val) * r i ≤
μ (σ (i.val+1)) * (k i.val (σ (i.val+1)) (m i.val) * a i (σ 0) (m' i) * v i)) :
μ (σ 0) * ∏ i, r i ≤
joint μ k u a n (σ n,(fun i => m i.val),m') * ∏ i, v i := by
let r' : ℕ → ℝ := fun j => if hj : j<n then r ⟨j,hj⟩ else 0
let v' : ℕ → ℝ := fun j => if hj : j<n then
k j (σ (j+1)) (m j) * a ⟨j,hj⟩ (σ 0) (m' ⟨j,hj⟩) * v ⟨j,hj⟩ else 0
have hh := trajectory_dominance (fun j => μ (σ j)) r' v' n
(fun j hj => by simpa only [r',dite_eq_left hj] using hr ⟨j,hj⟩)
(fun j hj => by simpa only [v',dite_eq_left hj] using hv ⟨j,hj⟩)
(fun j hj => by simpa only [r',v',dite_eq_left hj] using hs ⟨j,hj⟩)
simp only [r',v',dite_eq_left (Fin.isLt _),Finset.prod_mul_distrib] at hh
unfold joint terminalWeight
dsimp only
rw [recover_trajectory u σ m n hu,
weight_trajectory k u σ m (fun j => k j (σ (j+1)) (m j)) n hu (fun _ _ => rfl)]
convert hh using 1
ring
end CriticalPercolation.ReverseReference
end
end
end OAI
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