openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBlockConvergence.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockError | |
| namespace OAI | |
| /-! Actual equivalence of constrained and canonical block values along | |
| arbitrarily deep field partitions in a fixed bounded-height region. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped BigOperators NNReal Topology | |
| namespace InvariantIsing | |
| theorem magneticBlockGap_tendsto {A : Type*} [Fintype A] [DecidableEq A] | |
| (N : ℕ → ℕ) (hN : ∀ n, 0 < N n) (hNlim : Tendsto N atTop atTop) | |
| (group : ∀ n, Fin (N n) → A) (k : ℕ → A → ℕ) | |
| (hk : ∀ n a, k n a ≤ spinGroupSize (group n) a) (m : ℕ → A → ℝ) | |
| {r H : ℝ} (hr : r < 1) (hm : ∀ n a, |m n a| ≤ r) | |
| (hc : ∀ n a, (k n a : ℝ) = spinGroupSize (group n) a * ((1 + m n a) / 2)) | |
| (h : ℕ → FieldStep) (hH : ∀ n, (h n).height (Fin.last (h n).depth) ≤ H) : | |
| Tendsto (fun n => (N n : ℝ)⁻¹ * (∑ i, constrainedFieldValue (h n) (m n (group n i))) - | |
| constrainedBlockValue (spinGroupSlice (group n) (k n)) (h n)) atTop (𝓝 0) := by | |
| let B := magneticUniformBiasBound H r | |
| let C := 2 * Real.exp (2 * B + 4 * H) | |
| have hC : 0 < C := mul_pos (by norm_num) (Real.exp_pos _) | |
| have hB (n : ℕ) (i : Fin (N n)) : |magneticBias (h n) (m n (group n i))| ≤ B := | |
| magneticBias_uniform_bound (h n) hr (hm n _) (hH n) | |
| have hmi (n : ℕ) (a : A) : |m n a| < 1 := (hm n a).trans_lt hr | |
| let f := fun n => (N n : ℝ)⁻¹ * (∑ i, constrainedFieldValue (h n) (m n (group n i))) - | |
| constrainedBlockValue (spinGroupSlice (group n) (k n)) (h n) | |
| have hf (n : ℕ) : 0 ≤ f n := | |
| (magneticBlockGap_estimate (hN n) (group n) (k n) (hk n) (m n) | |
| (hmi n) (hc n) (h n) (hB n) (hH n) (by norm_num : (0 : ℝ) < 1) | |
| (by norm_num : (0 : ℝ) ≤ 0)).1 | |
| apply tendsto_zero_of_magneticBlockError N hNlim f hf C (Fintype.card A) hC | |
| intro ε hε t ht n | |
| have he := (magneticBlockGap_estimate (hN n) (group n) (k n) (hk n) (m n) | |
| (hmi n) (hc n) (h n) (hB n) (hH n) hε ht).2 | |
| rw [normalized_magneticBlockError (hN n)] at he | |
| exact he | |
| end InvariantIsing | |
| end | |
| end OAI | |