| import OAI.Probability.InvariantIsing.Arrays.PerturbationWeights |
|
|
| namespace OAI |
|
|
| /-! A concrete complete enumeration of the spectral/tree monomials with linear degree cap. -/ |
|
|
| noncomputable section |
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|
| open scoped BigOperators |
|
|
| namespace InvariantIsing |
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|
| def multidegreeDecode : (m : ℕ) → ℕ → Fin m → ℕ |
| | 0, _ => Fin.elim0 |
| | m + 1, j => Fin.cases (Nat.unpair j).1 (multidegreeDecode m (Nat.unpair j).2) |
|
|
| def multidegreeEncode : (m : ℕ) → (Fin m → ℕ) → ℕ |
| | 0, _ => 0 |
| | m + 1, d => Nat.pair (d 0) (multidegreeEncode m (Fin.tail d)) |
|
|
| lemma multidegreeDecode_encode (m : ℕ) (d : Fin m → ℕ) : |
| multidegreeDecode m (multidegreeEncode m d) = d := by |
| induction m with |
| | zero => exact Subsingleton.elim _ _ |
| | succ m ih => |
| funext i |
| induction i using Fin.cases with |
| | zero => simp only [multidegreeDecode, multidegreeEncode, Nat.unpair_pair, Fin.cases_zero] |
| | succ i => |
| simpa only [multidegreeDecode, multidegreeEncode, Nat.unpair_pair, Fin.cases_succ, |
| Fin.tail] using congrFun (ih (Fin.tail d)) i |
|
|
| lemma multidegreeDecode_sum_le (m j : ℕ) : |
| (∑ i, multidegreeDecode m j i) ≤ j := by |
| induction m generalizing j with |
| | zero => simp |
| | succ m ih => |
| rw [Fin.sum_univ_succ] |
| simp only [multidegreeDecode, Fin.cases_zero, Fin.cases_succ] |
| exact (Nat.add_le_add_left (ih (Nat.unpair j).2) _).trans (Nat.unpair_add_le j) |
|
|
| /-- First `m` coordinates are spectral degrees; the final coordinate is |
| the tree degree. The constant kernel can be retained harmlessly. -/ |
| def enumeratedSpectralDegree (m j : ℕ) (a : Fin m) : ℕ := |
| multidegreeDecode (m + 1) (j + 1) a.castSucc |
|
|
| def enumeratedTreeDegree (m j : ℕ) : ℕ := |
| multidegreeDecode (m + 1) (j + 1) (Fin.last m) |
|
|
| lemma enumeratedMonomial_degree_le (m j : ℕ) : |
| (∑ a, enumeratedSpectralDegree m j a) + enumeratedTreeDegree m j ≤ j + 1 := by |
| simpa only [Fin.sum_univ_castSucc, enumeratedSpectralDegree, enumeratedTreeDegree] using |
| multidegreeDecode_sum_le (m + 1) (j + 1) |
|
|
| lemma enumeratedSpectralDegree_real_sum_le (m j : ℕ) : |
| (∑ a, (enumeratedSpectralDegree m j a : ℝ)) ≤ (j : ℝ) + 1 := by |
| have h := (Nat.le_add_right (∑ a, enumeratedSpectralDegree m j a) |
| (enumeratedTreeDegree m j)).trans (enumeratedMonomial_degree_le m j) |
| exact_mod_cast h |
|
|
| /-- Every nonconstant joint spectral/tree monomial occurs at a fixed |
| finite index, hence eventually occurs in the truncation through `N`. -/ |
| theorem enumeratedMonomial_surjective (m : ℕ) (d : Fin m → ℕ) (r : ℕ) |
| (h : 1 ≤ (∑ a, d a) + r) : |
| ∃ j, (∀ a, enumeratedSpectralDegree m j a = d a) ∧ enumeratedTreeDegree m j = r := by |
| let v : Fin (m + 1) → ℕ := Fin.lastCases r d |
| let c := multidegreeEncode (m + 1) v |
| have hc : 1 ≤ c := by |
| have hs := multidegreeDecode_sum_le (m + 1) c |
| rw [multidegreeDecode_encode] at hs |
| rw [Fin.sum_univ_castSucc] at hs |
| simp only [v, Fin.lastCases_castSucc, Fin.lastCases_last] at hs |
| exact h.trans hs |
| refine ⟨c - 1, ?_, ?_⟩ |
| · intro a |
| have he := congrFun (multidegreeDecode_encode (m + 1) v) a.castSucc |
| simpa only [enumeratedSpectralDegree, Nat.sub_add_cancel hc, v, Fin.lastCases_castSucc] using he |
| · have he := congrFun (multidegreeDecode_encode (m + 1) v) (Fin.last m) |
| simpa only [enumeratedTreeDegree, Nat.sub_add_cancel hc, v, Fin.lastCases_last] using he |
|
|
| end InvariantIsing |
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| end |
|
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| end OAI |
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|
