| import Mathlib |
| import OAI.Probability.CriticalPercolation.Entropy.BinaryConfiguration |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace CriticalPercolation.BinaryLaw |
| open scoped Classical |
| open FiniteProduct BondGraph BinaryInterpolation ExactExposure FiniteEntropy |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
| local instance costBoundFinDec (n : ℕ) : DecidableEq (Fin n) := Classical.decEq _ |
| variable (G : BondGraph V E) (order : List (LazyDFS.Dart V E)) (d : ℕ) (x : ℕ → V) |
| variable (p : ℝ) (w : E → Prop → ℝ) (s n : ℕ) (D : FixedDescriptor n) |
|
|
| lemma constructionCost_le {c : ℝ} (hc : 0<c) |
| (hδ : 0 < prob w (endpointEvent G (x 0) (x d) s)) |
| (hD : (8:ℝ)^(-(n:ℤ)) ≤ descriptorMass G order d x p w s n D) : |
| constructionCost G order d x p w c s n D ≤ |
| Real.log (1/prob w (endpointEvent G (x 0) (x d) s)) + n*(Real.log 8+Real.log (1/c)) := by |
| have ho : 0 < descriptorMass G order d x p w s n D := |
| (zpow_pos (by norm_num : (0:ℝ)<8) _).trans_le hD |
| have hl := Real.log_le_log (zpow_pos (by norm_num : (0:ℝ)<8) _) hD |
| rw [Real.log_zpow] at hl |
| simp only [Int.cast_neg,Int.cast_natCast] at hl |
| unfold constructionCost |
| rw [Real.log_div (mul_ne_zero (inv_ne_zero hδ.ne') (pow_ne_zero _ (inv_ne_zero hc.ne'))) ho.ne', |
| Real.log_mul (inv_ne_zero hδ.ne') (pow_ne_zero _ (inv_ne_zero hc.ne')),Real.log_pow] |
| simp only [one_div] |
| nlinarith |
|
|
| end CriticalPercolation.BinaryLaw |
| end |
| end |
| end |
|
|
| end OAI |
| |
