File size: 3,504 Bytes
c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Cavity.CavityCanonicalMass
import Mathlib.Algebra.Order.Floor.Semiring
namespace OAI
/-! Positive rational approximations to a finite probability vector,
with explicit integer multiplicities. -/
noncomputable section
open Filter
open scoped Topology BigOperators
namespace InvariantIsing
def positiveApproxCount (x : ℝ) (k : ℕ) : ℕ := ⌊(k+1 : ℝ)*x⌋₊+1
lemma positiveApproxCount_pos (x : ℝ) (k : ℕ) : 0 < positiveApproxCount x k := by
unfold positiveApproxCount
omega
lemma positiveApproxCount_ratio (x : ℝ) (hx : 0 ≤ x) :
Tendsto (fun k => (positiveApproxCount x k : ℝ)/(k+1)) atTop (𝓝 x) := by
have herror k : 0 ≤ (positiveApproxCount x k : ℝ)/(k+1)-x ∧
(positiveApproxCount x k : ℝ)/(k+1)-x ≤ 1/(k+1) := by
have hk : 0 < (k+1 : ℝ) := by positivity
have hlow := Nat.lt_floor_add_one ((k+1 : ℝ)*x)
have hupp := Nat.floor_le (mul_nonneg hk.le hx)
have heq : (positiveApproxCount x k : ℝ)/(k+1)-x=
(((⌊(k+1 : ℝ)*x⌋₊ : ℕ) : ℝ)+1-x*(k+1))/(k+1) := by
simp only [positiveApproxCount,Nat.cast_add,Nat.cast_one]
field_simp
rw [heq]
constructor
· apply div_nonneg _ hk.le
nlinarith
· apply div_le_div_of_nonneg_right _ hk.le
nlinarith
have he := squeeze_zero (fun k => (herror k).1) (fun k => (herror k).2)
(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
simpa only [sub_add_cancel,zero_add] using he.add_const x
def positiveApproxDenominator {m : ℕ} (ρ : Fin m → ℝ) (k : ℕ) : ℕ :=
∑ a, positiveApproxCount (ρ a) k
lemma positiveApproxDenominator_pos {m : ℕ} (hm : 0 < m) (ρ : Fin m → ℝ) (k : ℕ) :
0 < positiveApproxDenominator ρ k := by
have hle := Finset.single_le_sum (f := fun a => positiveApproxCount (ρ a) k)
(fun _ _ => Nat.zero_le _) (Finset.mem_univ (⟨0,hm⟩ : Fin m))
exact (positiveApproxCount_pos (ρ ⟨0,hm⟩) k).trans_le hle
lemma positiveApproxWeights_tendsto {m : ℕ} (ρ : Fin m → ℝ)
(hρ : ∀ a, 0 ≤ ρ a) (hsum : ∑ a, ρ a=1) :
Tendsto (fun k a => (positiveApproxCount (ρ a) k : ℝ)/positiveApproxDenominator ρ k)
atTop (𝓝 ρ) := by
have hden : Tendsto (fun k => (positiveApproxDenominator ρ k : ℝ)/(k+1)) atTop (𝓝 (1:ℝ)) := by
have hh := tendsto_finsetSum Finset.univ (fun a _ => positiveApproxCount_ratio (ρ a) (hρ a))
simpa only [positiveApproxDenominator,Nat.cast_sum,Finset.sum_div,hsum] using hh
apply tendsto_pi_nhds.mpr
intro a
have hh := (positiveApproxCount_ratio (ρ a) (hρ a)).div hden one_ne_zero
simp only [div_one] at hh
apply hh.congr
intro k
exact div_div_div_cancel_right₀ (by positivity : (k+1 : ℝ)≠0) _ _
def positiveApproxWeights {m : ℕ} (ρ : Fin m → ℝ) (k : ℕ) (a : Fin m) : ℝ :=
(positiveApproxCount (ρ a) k : ℝ)/positiveApproxDenominator ρ k
lemma positiveApproxWeights_pos {m : ℕ} (hm : 0 < m) (ρ : Fin m → ℝ) (k : ℕ) (a : Fin m) :
0 < positiveApproxWeights ρ k a :=
div_pos (by exact_mod_cast positiveApproxCount_pos (ρ a) k)
(by exact_mod_cast positiveApproxDenominator_pos hm ρ k)
lemma positiveApproxWeights_sum {m : ℕ} (hm : 0 < m) (ρ : Fin m → ℝ) (k : ℕ) :
∑ a, positiveApproxWeights ρ k a=1 := by
simp only [positiveApproxWeights,← Finset.sum_div,← Nat.cast_sum]
exact div_self (Nat.cast_ne_zero.mpr (positiveApproxDenominator_pos hm ρ k).ne')
end InvariantIsing
end
end OAI
|