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import OAI.Probability.LogConcave.JetEstimates.Bound
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
variable {I J K L : Type*} [Fintype I] [Fintype J] [Fintype K] [Fintype L]
lemma Bound.comp {A : I → J → ℝ} {B : J → K → ℝ} {C D : ℝ}
(hA : Bound A C) (hB : Bound B D) (hC : 0≤C) :
Bound (fun i k => ∑ j,A i j*B j k) (C*D) := by
intro v
have he (i : I) : (∑ k,(∑ j,A i j*B j k)*v k)=
∑ j,A i j*(∑ k,B j k*v k) := by
simp only [Finset.sum_mul,Finset.mul_sum,mul_assoc]
exact Finset.sum_comm
simp_rw [he]
exact (hA _).trans ((mul_le_mul_of_nonneg_left (hB v) hC).trans_eq (by ring))
lemma bound_identity [DecidableEq I] : Bound (fun i j : I => if i=j then (1:ℝ) else 0) 1 := by
intro v
simp
lemma Bound.contract {P : Type*} [Fintype P]
{A : I → J × P → ℝ} {B : P × K → L → ℝ} {C D : ℝ}
(hA : Bound A C) (hB : Bound B D) (hC : 0≤C) :
Bound (fun (i : I × K) (j : J × L) => ∑ p,A i.1 (j.1,p)*B (p,i.2) j.2) (C*D) := by
classical
have h₁ := hA.kronecker (bound_identity (I:=K)) hC
have h₂ := (bound_identity (I:=J)).kronecker hB (by norm_num)
have h₂' := h₂.reindex (Equiv.prodAssoc J P K) (Equiv.refl (J×L))
have hh := h₁.comp h₂' (by simpa using hC)
simp only [mul_one,one_mul] at hh
convert! hh using 1
funext i j
simp only [Fintype.sum_prod_type,Equiv.prodAssoc_apply,Equiv.refl_apply]
simp [mul_ite,ite_mul]
lemma Bound.entry_sq {A : I → J → ℝ} {C : ℝ} (hA : Bound A C) (i : I) (j : J) :
(A i j)^2 ≤ C := by
classical
have hh := hA (fun k => if k=j then 1 else 0)
simp only [mul_ite,mul_one,mul_zero,Finset.sum_ite_eq',Finset.mem_univ,ite_true,
ite_pow,one_pow,zero_pow (by decide : (2:ℕ)≠0)] at hh
exact (Finset.single_le_sum (fun k _ => sq_nonneg (A k j)) (Finset.mem_univ i)).trans hh
lemma Bound.trace_sq {A : I → I → ℝ} {C : ℝ} (hA : Bound A C) :
(∑ i,A i i)^2 ≤ (Fintype.card I:ℝ)^2*C := by
have hh := dot_sq (fun _ : I => (1:ℝ)) (fun i => A i i)
simp only [one_mul,one_pow,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] at hh
have hg := Finset.sum_le_sum (fun i (_ : i∈(Finset.univ:Finset I)) => hA.entry_sq i i)
simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] at hg
exact hh.trans ((mul_le_mul_of_nonneg_left hg (Nat.cast_nonneg _)).trans_eq (by ring))
end LogConcaveSampling.TensorEnergy
namespace LogConcaveSampling.TensorEnergy
open scoped BigOperators
def wireStep {I O S X Y : Type*} [DecidableEq S]
(A : O → I → ℝ) (ei : X ≃ I×S) (eo : Y ≃ O×S) : Y → X → ℝ :=
fun y x => A (eo y).1 (ei x).1 * if (eo y).2=(ei x).2 then 1 else 0
lemma Bound.wireStep {I O S X Y : Type*} [Fintype I] [Fintype O] [Fintype S]
[Fintype X] [Fintype Y] [DecidableEq S] {A : O → I → ℝ} {C : ℝ}
(hA : Bound A C) (hC : 0≤C) (ei : X ≃ I×S) (eo : Y ≃ O×S) :
Bound (wireStep A ei eo) C := by
have hh := (hA.kronecker (bound_identity (I:=S)) hC).reindex eo ei
change Bound (fun y x => A (eo y).1 (ei x).1 * if (eo y).2=(ei x).2 then 1 else 0) C
simpa only [mul_one] using hh
def circuit {D : ℕ → Type*} [∀k,Fintype (D k)] [DecidableEq (D 0)]
(A : ∀k,D (k+1) → D k → ℝ) : ∀n,D n → D 0 → ℝ
| 0 => fun i j => if i=j then 1 else 0
| n+1 => fun i j => ∑k,A n i k*circuit A n k j
lemma circuit_bound {D : ℕ → Type*} [∀k,Fintype (D k)] [DecidableEq (D 0)]
(A : ∀k,D (k+1) → D k → ℝ) (C : ℕ → ℝ)
(hC : ∀k,0≤C k) (hA : ∀k,Bound (A k) (C k)) (n : ℕ) :
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 =>
simpa only [circuit,Finset.prod_range_succ,mul_comm] using (hA n).comp ih (hC n)
lemma Bound.trace_abs {I : Type*} [Fintype I] {A : I → I → ℝ} {C : ℝ}
(hA : Bound A (C^2)) (hC : 0≤C) : |∑i,A i i|≤(Fintype.card I:ℝ)*C := by
have hh := hA.trace_sq
apply (sq_le_sq₀ (abs_nonneg _) (mul_nonneg (Nat.cast_nonneg _) hC)).mp
simpa only [sq_abs,mul_pow] using hh
theorem closed_wire_circuit_bound
{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)
(hC : ∀k,0≤C k) (hA : ∀k,Bound (A k) ((C k)^2))
(ei : ∀k,D k ≃ I k×S k) (eo : ∀k,D (k+1) ≃ O k×S k)
(n : ℕ) (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) : Bound (wireStep (A k) (ei k) (eo k)) ((C k)^2) :=
(hA k).wireStep (sq_nonneg _) (ei k) (eo k)
have hh := (circuit_bound _ (fun k => (C k)^2) (fun k => sq_nonneg _) hlocal n).reindex
close (Equiv.refl (D 0))
rw [Finset.prod_pow] at hh
exact hh.trace_abs (Finset.prod_nonneg (fun k _ => hC k))
lemma cut_bundle_card (d b : ℕ) : Fintype.card (Fin b → Fin d)=d^b := by simp
end LogConcaveSampling.TensorEnergy
namespace LogConcaveSampling.TensorEnergy
open scoped BigOperators Classical
variable {D : ℕ → Type*} [∀k,Fintype (D k)] [DecidableEq (D 0)]
abbrev Path (n : ℕ) := ∀k : Fin (n+1),D k.val
def pathWeight (A : ∀k,D (k+1) → D k → ℝ) (n : ℕ) (h : Path (D:=D) n) : ℝ :=
∏k : Fin n,A k.val (h k.succ) (h k.castSucc)
omit [∀k,Fintype (D k)] [DecidableEq (D 0)] in
lemma pathWeight_snoc (A : ∀k,D (k+1) → D k → ℝ) (n : ℕ)
(h : Path (D:=D) n) (y : D (n+1)) :
pathWeight A (n+1) (Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y)=pathWeight A n h*A n y (h (Fin.last n)) := by
unfold pathWeight
rw [Fin.prod_univ_castSucc]
congr 1
· apply Finset.prod_congr rfl
intro k hk
change A k.val (Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y k.succ.castSucc)
(Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y k.castSucc.castSucc) = _
simp only [Fin.snoc_castSucc]
· simp only [Fin.succ_last,Fin.snoc_last,Fin.snoc_castSucc,Fin.val_last]
theorem circuit_path_expansion (A : ∀k,D (k+1) → D k → ℝ) (n : ℕ)
(B : D 0 → D n → ℝ) :
(∑h : Path (D:=D) n,pathWeight A n h*B (h 0) (h (Fin.last n)))=
∑i,∑j,circuit A n j i*B i j := by
induction n with
| zero =>
rw [←(Fin.consEquiv (fun k : Fin 1 => D k.val)).sum_comp]
simp [pathWeight,circuit,Fintype.sum_prod_type]
| succ n ih =>
rw [←(Fin.snocEquiv (fun k : Fin (n+2) => D k.val)).sum_comp]
simp only [Fintype.sum_prod_type,Fin.snocEquiv_apply]
change (∑y : D (n+1),∑h : Path (D:=D) n,
pathWeight A (n+1) (Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y)*
B (Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y (0 : Fin (n+1)).castSucc)
(Fin.snoc (α:=fun k : Fin (n+2) => D k.val) h y (Fin.last (n+1))))=_
simp_rw [pathWeight_snoc,Fin.snoc_castSucc,Fin.snoc_last]
have hh (y : D (n+1)) :
(∑h : Path (D:=D) n,pathWeight A n h*A n y (h (Fin.last n))*B (h 0) y)=
∑i,∑j,circuit A n j i*(A n y j*B i y) := by
simpa only [mul_assoc] using ih (fun i j => A n y j*B i y)
simp_rw [hh]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro y hy
simp only [circuit,Finset.sum_mul]
apply Finset.sum_congr rfl
intro j hj
ring
end LogConcaveSampling.TensorEnergy
end UpperProof
end
end
end
end OAI
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