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import OAI.Analysis.HardSphere.Geometry.EnnrealTsumFinset
import OAI.Analysis.HardSphere.Geometry.NominalDrift
namespace OAI
noncomputable section
open scoped BigOperators Classical
open scoped BigOperators Classical
open scoped BigOperators Classical
open scoped ENNReal NNReal
namespace HardSphere.Lattice
lemma geometricEnvelope_ne_top (A R : ℕ) (b q : ℝ≥0∞) (hb : b≠⊤) (hq : q<1)
(z : ℤ×ℤ) : radiusEnvelope (geometricRadiusWeights A R b q) z≠⊤ := by
rw [geometricRadiusWeights_envelope]
apply ENNReal.add_ne_top.mpr
constructor
· split_ifs <;> simp_all
· apply ne_top_of_le_ne_top (b:=∑' m : ℕ, q^(m+1))
· have he : (∑' m : ℕ, q^(m+1))=q*(∑' m : ℕ,q^m) := by
simp_rw [pow_succ',ENNReal.tsum_mul_left]
rw [he,ENNReal.tsum_geometric]
exact ENNReal.mul_ne_top (ne_of_lt (hq.trans (by norm_num)))
(ENNReal.inv_ne_top.mpr (ne_of_gt (tsub_pos_iff_lt.mpr hq)))
· exact ENNReal.tsum_le_tsum fun m ↦ by split_ifs <;> simp
def candidateConstant (a : ℕ) : ℕ := (2*(a+3)+1)^3*2^((2*a+1)^3)+(12*a+24)
def encounterConstant (R : ℕ) : ℕ := 343*(2*R+1)^3
lemma candidateConstant_growth (a : ℕ) :
(2*(a+3)+1)^3*2^((2*a+1)^3)≤candidateConstant a := Nat.le_add_right _ _
lemma candidateConstant_radius (a : ℕ) : 12*a+24≤candidateConstant a := Nat.le_add_left _ _
lemma encounterConstant_detour (R : ℕ) : 52*(2*R+1)^3≤encounterConstant R := by
exact Nat.mul_le_mul_right _ (by norm_num)
theorem nominal_potential_bound {K m n : ℕ} [NeZero K] (R : ℕ)
(hn : 0<n) (hnK : 2*n≤K) (ht : n/2≤3^m)
(v : Vertex) (Δ₂ Δ₃ : ℤ) (hΔ₂ : |Δ₂|≤2*n) (hΔ₃ : |Δ₃|≤2*n)
(w : FourNoise m) (κ t : ℝ) (hκ : 0≤κ) (ht0 : 0≤t)
(hqt : (candidateConstant (R+2):ℝ)*t<1) :
let U := nominalVertices m n v Δ₂ Δ₃ (firstPathNoise w)
let U' := nominalVertices m n v Δ₂ Δ₃ (secondPathNoise w)
let Γ := (nominalCover U).image (torusProjection K)
let Γ' := (nominalCover U').image (torusProjection K)
let weights := geometricRadiusWeights (candidateConstant (R+2)) (4*(R+6))
(ENNReal.ofReal ((encounterConstant R:ℝ)*κ))
(ENNReal.ofReal ((candidateConstant (R+2):ℝ)*t))
κ*((encounterSet (torusNear K R) Γ Γ').card:ℝ)+
(∑ F∈candidateSets (torusRangeGraph K (R+2)),
if torusContact (R+2) F Γ ∧ torusContact (R+2) F Γ' then t^F.card else 0) ≤
∑ j∈Finset.range (n+1), radiusPotential weights (differenceAt m (min j (n-j)) w) := by
classical
dsimp only
let U := nominalVertices m n v Δ₂ Δ₃ (firstPathNoise w)
let U' := nominalVertices m n v Δ₂ Δ₃ (secondPathNoise w)
let Γ := (nominalCover U).image (torusProjection K)
let Γ' := (nominalCover U').image (torusProjection K)
let A := candidateConstant (R+2)
let C := encounterConstant R
let weights := geometricRadiusWeights A (4*(R+6)) (ENNReal.ofReal ((C:ℝ)*κ))
(ENNReal.ofReal ((A:ℝ)*t))
let z (i : Fin (n+1)) := differenceAt m (min i.val (n-i.val)) w
have hspeed := nominal_speed m n hn ht v Δ₂ Δ₃ hΔ₂ hΔ₃ (secondPathNoise w)
have hfirst := nominal_first m n v Δ₂ Δ₃ (firstPathNoise w)
have hfirst' := nominal_first m n v Δ₂ Δ₃ (secondPathNoise w)
have hnear (r : ℕ) (i : Fin (n+1)) :
torusNear K r (torusProjection K (U i)) (torusProjection K (U' i)) ↔ z i∈square r :=
nominal_near_iff m r hnK ht v Δ₂ Δ₃ w i
have hN := nominal_encounter_bound R hnK U U' (v 0) hfirst hfirst' hspeed Γ Γ'
(fun _ hx ↦ nominalCover_projection U hx) (fun _ hx ↦ nominalCover_projection U' hx)
have hNen : ENNReal.ofReal (κ*((encounterSet (torusNear K R) Γ Γ').card:ℝ))≤
∑ i : Fin (n+1), if z i∈square (4*(R+6)) then ENNReal.ofReal ((C:ℝ)*κ) else 0 := by
calc
_ ≤ ENNReal.ofReal ((C:ℝ)*κ*
((Finset.univ.filter (fun i ↦ torusNear K (4*(R+6))
(torusProjection K (U i)) (torusProjection K (U' i)))).card:ℝ)) := by
apply ENNReal.ofReal_le_ofReal
have hN' : ((encounterSet (torusNear K R) Γ Γ').card:ℝ)≤
(C:ℝ)*((Finset.univ.filter (fun i ↦ torusNear K (4*(R+6))
(torusProjection K (U i)) (torusProjection K (U' i)))).card:ℝ) := by exact_mod_cast hN
nlinarith
_ = _ := by
rw [ENNReal.ofReal_mul (mul_nonneg (Nat.cast_nonneg _) hκ),ENNReal.ofReal_natCast]
simp_rw [← hnear]
rw [← Finset.sum_filter]
simp only [Finset.sum_const,nsmul_eq_mul]
rw [mul_comm]
have hF := contact_candidate_potential (R+2) hnK U U' (v 0) hfirst hfirst' hspeed Γ Γ'
(fun _ hx ↦ nominalCover_projection U hx) (fun _ hx ↦ nominalCover_projection U' hx) (ENNReal.ofReal t)
have hFen : ENNReal.ofReal (∑ F∈candidateSets (torusRangeGraph K (R+2)),
if torusContact (R+2) F Γ ∧ torusContact (R+2) F Γ' then t^F.card else 0)≤
∑ i : Fin (n+1), ∑' l : ℕ, if z i∈square (A*(l+1))
then (ENNReal.ofReal ((A:ℝ)*t))^(l+1) else 0 := by
have hcast : ENNReal.ofReal (∑ F∈candidateSets (torusRangeGraph K (R+2)),
if torusContact (R+2) F Γ ∧ torusContact (R+2) F Γ' then t^F.card else 0)=
∑ F∈candidateSets (torusRangeGraph K (R+2)),
if torusContact (R+2) F Γ ∧ torusContact (R+2) F Γ' then (ENNReal.ofReal t)^F.card else 0 := by
rw [ENNReal.ofReal_sum_of_nonneg (by intro F _; split_ifs <;> positivity)]
apply Finset.sum_congr rfl
intro F _
split_ifs <;> simp [ENNReal.ofReal_pow ht0]
rw [hcast]
refine hF.trans (Finset.sum_le_sum fun i _ ↦ ENNReal.tsum_le_tsum fun l ↦ ?_)
by_cases h : torusNear K ((12*(R+2)+24)*(l+1)) (torusProjection K (U i)) (torusProjection K (U' i))
· have hz : z i∈square (A*(l+1)) := (hnear _ i).mp
(torusNear_mono (Nat.mul_le_mul_right _ (candidateConstant_radius (R+2))) h)
rw [ite_eq_left h,ite_eq_left hz]
apply pow_le_pow_left'
rw [ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast]
apply mul_le_mul' _ le_rfl
exact_mod_cast candidateConstant_growth (R+2)
· rw [ite_eq_right h]; exact zero_le
have hfinite (i : Fin (n+1)) : radiusEnvelope weights (z i)≠⊤ := by
apply geometricEnvelope_ne_top _ _ _ _ ENNReal.ofReal_ne_top
exact (ENNReal.ofReal_lt_one).mpr hqt
have hbound := add_le_add hNen hFen
rw [← Finset.sum_add_distrib] at hbound
have hrw (i : Fin (n+1)) :
(if z i∈square (4*(R+6)) then ENNReal.ofReal ((C:ℝ)*κ) else 0)+
(∑' l : ℕ, if z i∈square (A*(l+1)) then (ENNReal.ofReal ((A:ℝ)*t))^(l+1) else 0)=
ENNReal.ofReal (radiusPotential weights (z i)) := by
rw [radiusPotential,ENNReal.ofReal_toReal (hfinite i)]
exact (geometricRadiusWeights_envelope _ _ _ _ _).symm
simp_rw [hrw] at hbound
rw [← ENNReal.ofReal_sum_of_nonneg (fun i _ ↦ radiusPotential_nonneg weights (z i))] at hbound
rw [← ENNReal.ofReal_add (mul_nonneg hκ (Nat.cast_nonneg _))
(Finset.sum_nonneg (by intro F _; split_ifs <;> positivity))] at hbound
have hreal := (ENNReal.ofReal_le_ofReal_iff (Finset.sum_nonneg
(fun i _ ↦ radiusPotential_nonneg weights (z i)))).mp hbound
dsimp only [z] at hreal
rw [Fin.sum_univ_eq_sum_range (fun j ↦ radiusPotential weights (differenceAt m (min j (n-j)) w)) (n+1)] at hreal
exact hreal
end HardSphere.Lattice
open scoped ENNReal
namespace HardSphere.Lattice
universe u
variable {V : Type u} [Fintype V] [DecidableEq V]
omit [Fintype V] in
lemma prod_one_add_ofReal_le_exp_sum {I : Type*} (Q : Finset I) (w : I → ℝ)
(hw : ∀ i ∈ Q, 0 ≤ w i) :
∏ i ∈ Q, (1 + ENNReal.ofReal (w i)) ≤
ENNReal.ofReal (Real.exp (∑ i ∈ Q, w i)) := by
classical
induction Q using Finset.induction_on with
| empty => simp
| @insert i Q hi ih =>
rw [Finset.prod_insert hi, Finset.sum_insert hi, Real.exp_add,
ENNReal.ofReal_mul (Real.exp_pos _).le]
have hwi := hw i (Finset.mem_insert_self i Q)
have hQ : ∀ j ∈ Q, 0 ≤ w j := fun j hj ↦ hw j (Finset.mem_insert_of_mem hj)
apply mul_le_mul' _ (ih hQ)
rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_add (by norm_num) hwi]
exact ENNReal.ofReal_le_ofReal (by simpa [add_comm] using Real.add_one_le_exp (w i))
theorem badComponents_exponential_bound (G : SimpleGraph V)
(μ : PMF (Finset V)) (p β : ℝ) (hp : 0 ≤ p)
(hbad : JointBadLaw μ (ENNReal.ofReal p)) (P : Finset V → Prop) :
(∑' B, μ B * ENNReal.ofReal (Real.exp
(β * ∑ F∈badComponents G B, if P F then (F.card:ℝ) else 0))) ≤
ENNReal.ofReal (Real.exp (∑ F∈candidateSets G,
if P F then (p * Real.exp β)^F.card else 0)) := by
let w : Finset V → ℝ≥0∞ := fun F ↦ if P F then
ENNReal.ofReal (Real.exp (β * (F.card:ℝ))) else 0
have hpoint (B : Finset V) :
ENNReal.ofReal (Real.exp
(β * ∑ F∈badComponents G B, if P F then (F.card:ℝ) else 0)) ≤
∏ F∈badComponents G B, (1+w F) := by
rw [Finset.mul_sum, Real.exp_sum]
rw [ENNReal.ofReal_prod_of_nonneg (fun _ _ ↦ (Real.exp_pos _).le)]
apply Finset.prod_le_prod
intro F _
by_cases h : P F
· simp only [ite_eq_left h,w]
exact le_add_self
· simp only [ite_eq_right h,w,mul_zero,Real.exp_zero,ENNReal.ofReal_one,add_zero]
exact le_rfl
calc
_ ≤ ∑' B, μ B * ∏ F∈badComponents G B, (1+w F) := by
exact ENNReal.tsum_le_tsum fun B ↦ mul_le_mul' le_rfl (hpoint B)
_ ≤ ∏ F∈candidateSets G, (1+(ENNReal.ofReal p)^F.card*w F) :=
environment_product_bound μ _ hbad (badComponents G)
(badComponents_disjoint G) (fun B F hF ↦ badComponents_subset G B hF)
(candidateSets G) (badComponents_candidate G) w
_ = ∏ F∈candidateSets G,
(1+ENNReal.ofReal (if P F then (p*Real.exp β)^F.card else 0)) := by
apply Finset.prod_congr rfl
intro F _
dsimp only [w]
by_cases h : P F
· simp only [ite_eq_left h]
rw [← ENNReal.ofReal_pow hp,← ENNReal.ofReal_mul (pow_nonneg hp _),
mul_comm β (F.card:ℝ),Real.exp_nat_mul,mul_pow]
· simp only [ite_eq_right h,mul_zero,ENNReal.ofReal_zero]
_ ≤ _ := prod_one_add_ofReal_le_exp_sum _ _ (by
intro F _
split_ifs
· exact pow_nonneg (mul_nonneg hp (Real.exp_pos _).le) _
· exact le_rfl)
end HardSphere.Lattice
end
end OAI
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