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dd14c26 | 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 | 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
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