File size: 3,415 Bytes
b317718 | 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 85 86 | import OAI.Probability.InvariantIsing.Arrays.PerturbationWeights
namespace OAI
/-! A concrete complete enumeration of the spectral/tree monomials with linear degree cap. -/
noncomputable section
open scoped BigOperators
namespace InvariantIsing
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
end
end OAI
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