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9b6e014 | 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 | import OAI.Probability.ContinuumTransition.ConstructionMasses
namespace OAI
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
lemma geometricBudget_tendsto : Tendsto geometricBudget atTop (𝓝 0) :=
tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
lemma ConstructionData.positions_cauchy {p : ℝ} (C : ConstructionData p) :
CauchySeq (fun n => (C.stages n).x) := by
apply cauchySeq_of_le_geometric (1/2) (1/32) (by norm_num)
intro n
have hh := (C.transitions n).displacement.le
rw [dist_comm] at hh
simpa only [geometricBudget,div_eq_mul_inv,mul_comm,one_mul] using hh
lemma ConstructionData.exists_limit_position {p : ℝ} (C : ConstructionData p) :
∃ x : Parameter, Tendsto (fun n => (C.stages n).x) atTop (𝓝 x) ∧
dist x initialPoint ≤ 1/8 := by
obtain ⟨x,hx⟩ := cauchySeq_tendsto_of_complete C.positions_cauchy
refine ⟨x,hx,?_⟩
apply le_of_tendsto (hx.dist tendsto_const_nhds)
exact Eventually.of_forall (fun n => (C.stages n).x_near.le)
lemma initial_near_beta {x : Parameter} (hx : dist x initialPoint ≤ 1/8) :
x.1 ∈ Ioo (1/2:ℝ) (3/2:ℝ) := by
have hh : |x.1-1| ≤ 1/8 ∧ |x.2-(-3/4)| ≤ 1/8 := by
simpa only [dist_eq_norm_sub,Prod.norm_def,Prod.fst_sub,Prod.snd_sub,
Real.norm_eq_abs,max_le_iff,initialPoint] using hx
have hb := hh.1
rcases abs_le.mp hb with ⟨h₁,h₂⟩
constructor <;> linarith
lemma initial_near_moderate {x : Parameter} (hx : dist x initialPoint ≤ 1/8) :
x ∈ moderateBox := initial_ball_moderate (hx.trans (by norm_num : (1:ℝ)/8 ≤ 1/2))
end ContinuumTemperature
end
end
end OAI
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