| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.WireStep |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| namespace LogConcaveSampling.TensorEnergy |
| open scoped BigOperators |
|
|
| lemma circuit_bound_upto {D : ℕ → Type*} [∀k,Fintype (D k)] [DecidableEq (D 0)] |
| (A : ∀k,D (k+1) → D k → ℝ) (C : ℕ → ℝ) (n : ℕ) |
| (hC : ∀k<n,0≤C k) (hA : ∀k<n,Bound (A k) (C k)) : |
| Bound (circuit A n) (∏k∈Finset.range n,C k) := by |
| induction n with |
| | zero => simpa only [circuit,Finset.range_zero,Finset.prod_empty] using (bound_identity (I:=D 0)) |
| | succ n ih => |
| have hi := ih (fun k hk => hC k (Nat.lt_trans hk (Nat.lt_succ_self n))) |
| (fun k hk => hA k (Nat.lt_trans hk (Nat.lt_succ_self n))) |
| simpa only [circuit,Finset.prod_range_succ,mul_comm] using |
| (hA n (Nat.lt_succ_self n)).comp hi (hC n (Nat.lt_succ_self n)) |
|
|
| theorem closed_wire_circuit_bound_upto |
| {D I O S : ℕ → Type*} [∀k,Fintype (D k)] [∀k,Fintype (I k)] |
| [∀k,Fintype (O k)] [∀k,Fintype (S k)] [∀k,DecidableEq (S k)] |
| [DecidableEq (D 0)] |
| (A : ∀k,O k → I k → ℝ) (C : ℕ → ℝ) |
| (ei : ∀k,D k ≃ I k×S k) (eo : ∀k,D (k+1) ≃ O k×S k) |
| (n : ℕ) (hC : ∀k<n,0≤C k) (hA : ∀k<n,Bound (A k) ((C k)^2)) |
| (close : D 0 ≃ D n) : |
| |∑i,circuit (fun k => wireStep (A k) (ei k) (eo k)) n (close i) i|≤ |
| (Fintype.card (D 0):ℝ)*(∏k∈Finset.range n,C k) := by |
| have hlocal (k : ℕ) (hk : k<n) : Bound (wireStep (A k) (ei k) (eo k)) ((C k)^2) := |
| (hA k hk).wireStep (sq_nonneg _) (ei k) (eo k) |
| have hh := (circuit_bound_upto _ (fun k => (C k)^2) n (fun k _ => sq_nonneg _) hlocal).reindex |
| close (Equiv.refl (D 0)) |
| rw [Finset.prod_pow] at hh |
| exact hh.trace_abs (Finset.prod_nonneg (fun k hk => hC k (Finset.mem_range.mp hk))) |
| end LogConcaveSampling.TensorEnergy |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
