openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Core /PositiveRationalApproximation.lean
| 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 | |