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import OAI.Combinatorics.GraphThreshold.ResamplingEvents
import OAI.Combinatorics.GraphThreshold.SpreadCounting
import OAI.Combinatorics.GraphThreshold.FiniteProbability
import OAI.Combinatorics.GraphThreshold.ResamplingDensity
namespace OAI
/-! # Local failure of the actual resampling experiment -/
noncomputable section
open Finset
namespace LeanBlast.SecondKahnKalai
variable {X B : Type*} [Fintype X] [DecidableEq X] [Fintype B]
omit [Fintype X] [Fintype B] in
private theorem target_subset_iff_required_subset (A : B → Finset X)
(W : Finset X) (c b : B) :
A b ⊆ resamplingY A W c ↔ A b \ A c ⊆ W := by
constructor
· intro h x hx
rcases Finset.mem_union.mp (h (Finset.mem_sdiff.mp hx).1) with hW | hc
· exact hW
· exact ((Finset.mem_sdiff.mp hx).2 hc).elim
· intro h x hx
by_cases hc : x ∈ A c
· exact Finset.mem_union.mpr (Or.inr hc)
· exact Finset.mem_union.mpr (Or.inl (h (Finset.mem_sdiff.mpr ⟨hx, hc⟩)))
omit [Fintype X] [Fintype B] in
private theorem fragment_subset_inter_of_target_subset (A : B → Finset X)
(W : Finset X) (c b : B) (h : A b ⊆ resamplingY A W c) :
A b \ W ⊆ A c ∩ A b := by
intro x hx
obtain ⟨hb, hW⟩ := Finset.mem_sdiff.mp hx
rcases Finset.mem_union.mp (h hb) with hxW | hc
· exact (hW hxW).elim
· exact Finset.mem_inter.mpr ⟨hc, hb⟩
omit [Fintype X] [Fintype B] in
private theorem inverse_power_card_cancel (A : B → Finset X) {ρ : ℝ}
(hρ : 0 < ρ) (c b : B) :
(ρ⁻¹) ^ (A b).card * ρ ^ (A b \ A c).card =
(ρ⁻¹) ^ (A c ∩ A b).card := by
have hc : (A b).card = (A b \ A c).card + (A c ∩ A b).card := by
simpa only [Finset.inter_comm] using (Finset.card_sdiff_add_card_inter (A b) (A c)).symm
rw [hc, pow_add]
calc
((ρ⁻¹) ^ (A b \ A c).card * (ρ⁻¹) ^ (A c ∩ A b).card) *
ρ ^ (A b \ A c).card =
((ρ⁻¹) ^ (A b \ A c).card * ρ ^ (A b \ A c).card) *
(ρ⁻¹) ^ (A c ∩ A b).card := by ring
_ = (ρ⁻¹) ^ (A c ∩ A b).card := by simp [inv_pow, hρ.ne']
omit [Fintype X] in
/-- At a large denominator, the actual large-fragment weight is bounded by
an intersection-tail indicator and the required-coordinate indicator. -/
theorem resampling_large_fragment_term_le (ν : FiniteLaw B) (A : B → Finset X)
{ρ : ℝ} (hρ : 0 < ρ) (m : ℕ) (W : Finset X) (c b : B) :
(if Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W c) ∧
m / 2 < (A b \ W).card then resamplingWeight ν A ρ W (c, b) else 0) ≤
if m < 2 * (A c ∩ A b).card ∧ A b \ A c ⊆ W then
Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b * (ρ⁻¹) ^ (A b).card
else 0 := by
classical
have hrhs : 0 ≤
if m < 2 * (A c ∩ A b).card ∧ A b \ A c ⊆ W then
Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b * (ρ⁻¹) ^ (A b).card
else 0 := by
split_ifs
· exact mul_nonneg (mul_nonneg (mul_nonneg (Real.exp_pos _).le (ν.nonneg c))
(ν.nonneg b)) (pow_nonneg (inv_nonneg.mpr hρ.le) _)
· exact le_rfl
by_cases he : Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W c) ∧
m / 2 < (A b \ W).card
· rw [ite_eq_left he]
by_cases hs : A b ⊆ resamplingY A W c
· have hfrag := Finset.card_le_card (fragment_subset_inter_of_target_subset A W c b hs)
have hlarge : m < 2 * (A c ∩ A b).card := by omega
have hreq := (target_subset_iff_required_subset A W c b).mp hs
rw [ite_eq_left ⟨hlarge, hreq⟩]
have hZpos : 0 < resamplingZ ν A ρ (resamplingY A W c) :=
(Real.exp_pos _).trans_le he.1
have hscale : 1 ≤ Real.exp (10 * (m : ℝ)) *
resamplingZ ν A ρ (resamplingY A W c) := by
calc
1 = Real.exp (10 * (m : ℝ)) * Real.exp (-10 * (m : ℝ)) := by
rw [← Real.exp_add]; simp
_ ≤ _ := mul_le_mul_of_nonneg_left he.1 (Real.exp_pos _).le
have hnum : 0 ≤ ν.weight b * (ρ⁻¹) ^ (A b).card :=
mul_nonneg (ν.nonneg b) (pow_nonneg (inv_nonneg.mpr hρ.le) _)
have hdiv : (ν.weight b * (ρ⁻¹) ^ (A b).card) /
resamplingZ ν A ρ (resamplingY A W c) ≤
Real.exp (10 * (m : ℝ)) * (ν.weight b * (ρ⁻¹) ^ (A b).card) := by
apply (div_le_iff₀ hZpos).mpr
nlinarith [mul_le_mul_of_nonneg_left hscale hnum]
unfold resamplingWeight resamplingTargetWeight
rw [resamplingNumerator, ite_eq_left hs]
calc
ν.weight c * ((ν.weight b * (ρ⁻¹) ^ (A b).card) /
resamplingZ ν A ρ (resamplingY A W c)) ≤
ν.weight c * (Real.exp (10 * (m : ℝ)) *
(ν.weight b * (ρ⁻¹) ^ (A b).card)) :=
mul_le_mul_of_nonneg_left hdiv (ν.nonneg c)
_ = _ := by ring
· rw [resamplingWeight_eq_zero_of_not_subset ν A ρ W c b hs]
exact hrhs
· rw [ite_eq_right he]
exact hrhs
private theorem local_bernoulliExpectation_sum {ι : Type*} [Fintype ι]
(ρ : ℝ) (f : ι → Finset X → ℝ) :
bernoulliExpectation ρ (fun W => ∑ i, f i W) =
∑ i, bernoulliExpectation ρ (f i) := by
simp only [bernoulliExpectation, Finset.mul_sum]
exact Finset.sum_comm
theorem resampling_large_fragment_pair_bound (ν : FiniteLaw B) (A : B → Finset X)
{ρ : ℝ} (hρ : 0 < ρ) (hρ1 : ρ < 1) (m : ℕ) (c b : B) :
bernoulliExpectation ρ (fun W =>
if Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W c) ∧
m / 2 < (A b \ W).card then resamplingWeight ν A ρ W (c, b) else 0) ≤
if m < 2 * (A c ∩ A b).card then
Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b * (ρ⁻¹) ^ (A c ∩ A b).card
else 0 := by
classical
let μ : FiniteLaw (Finset X) := bernoulliLaw ρ hρ.le hρ1.le
by_cases hlarge : m < 2 * (A c ∩ A b).card
· rw [ite_eq_left hlarge]
calc
_ ≤ μ.expectation (fun W =>
(Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b * (ρ⁻¹) ^ (A b).card) *
(if A b \ A c ⊆ W then 1 else 0)) := by
apply μ.expectation_mono
intro W
simpa only [hlarge, true_and, mul_ite, mul_one, mul_zero] using
resampling_large_fragment_term_le ν A hρ m W c b
_ = (Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b * (ρ⁻¹) ^ (A b).card) *
ρ ^ (A b \ A c).card := by
rw [FiniteLaw.expectation_smul]
congr 1
change bernoulliExpectation ρ (fun W => if A b \ A c ⊆ W then 1 else 0) = _
rw [← bernoulliProbability_subset ρ (A b \ A c)]
simp only [bernoulliExpectation, bernoulliProbability, mul_ite, mul_one, mul_zero]
apply Finset.sum_congr rfl
intro W _
by_cases hW : A b \ A c ⊆ W <;> simp [hW]
_ = Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b *
((ρ⁻¹) ^ (A b).card * ρ ^ (A b \ A c).card) := by ring
_ = _ := by rw [inverse_power_card_cancel A hρ c b]
· rw [ite_eq_right hlarge]
calc
_ ≤ μ.expectation (fun _ => 0) := by
apply μ.expectation_mono
intro W
simpa only [hlarge, false_and, ite_false] using
resampling_large_fragment_term_le ν A hρ m W c b
_ = 0 := μ.expectation_const 0
theorem resampling_large_fragment_bound (ν : FiniteLaw B) (A : B → Finset X)
{ρ a : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(m : ℕ) (hsize : ∀ b, (A b).card ≤ m) (hspread : IsSpread ν A a)
(ha : 0 ≤ a) (hratio : a / ρ ≤ Real.exp (-50)) :
bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).prob
(fun cb => Real.exp (-10 * (m : ℝ)) ≤
resamplingZ ν A ρ (resamplingY A W cb.1) ∧
m / 2 < (A cb.2 \ W).card)) ≤
Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) := by
classical
have hpair (b : B) :
(∑ c, bernoulliExpectation ρ (fun W =>
if Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W c) ∧
m / 2 < (A b \ W).card then resamplingWeight ν A ρ W (c, b) else 0)) ≤
(Real.exp (10 * (m : ℝ)) * ν.weight b) * largeIntersectionTail m (a / ρ) := by
calc
_ ≤ ∑ c, if m < 2 * (A c ∩ A b).card then
Real.exp (10 * (m : ℝ)) * ν.weight c * ν.weight b *
(ρ⁻¹) ^ (A c ∩ A b).card else 0 := by
apply Finset.sum_le_sum
intro c _
exact resampling_large_fragment_pair_bound ν A hρ hρ1 m c b
_ = (Real.exp (10 * (m : ℝ)) * ν.weight b) *
∑ c, if m < 2 * (A c ∩ A b).card then
ν.weight c * (ρ⁻¹) ^ (A c ∩ A b).card else 0 := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro c _
split_ifs <;> ring
_ ≤ _ := mul_le_mul_of_nonneg_left
(intersection_inverse_truncated_sum_le hspread ha (hsize b) hρ
(fun r => m < 2 * r))
(mul_nonneg (Real.exp_pos _).le (ν.nonneg b))
calc
_ = ∑ b, ∑ c, bernoulliExpectation ρ (fun W =>
if Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W c) ∧
m / 2 < (A b \ W).card then resamplingWeight ν A ρ W (c, b) else 0) := by
simp only [FiniteLaw.prob, resamplingLaw_weight, Fintype.sum_prod_type]
conv_lhs => arg 2; ext W; rw [Finset.sum_comm]
rw [local_bernoulliExpectation_sum]
simp_rw [local_bernoulliExpectation_sum]
apply Finset.sum_congr rfl
intro target _
apply Finset.sum_congr rfl
intro source _
congr 1
funext drawn
split_ifs <;> rfl
_ ≤ ∑ b, (Real.exp (10 * (m : ℝ)) * ν.weight b) * largeIntersectionTail m (a / ρ) :=
Finset.sum_le_sum (fun b _ => hpair b)
_ = Real.exp (10 * (m : ℝ)) * largeIntersectionTail m (a / ρ) := by
calc
_ = (Real.exp (10 * (m : ℝ)) * largeIntersectionTail m (a / ρ)) *
∑ b, ν.weight b := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro b _
ring
_ = _ := by rw [ν.total, mul_one]
_ ≤ Real.exp (10 * (m : ℝ)) * ((2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ))) :=
mul_le_mul_of_nonneg_left (largeIntersectionTail_le (div_nonneg ha hρ.le) hratio)
(Real.exp_pos _).le
_ = _ := by ring
/-- The density identity bounds the probability of a small denominator by
the cutoff itself. -/
theorem resampling_small_density_bound (ν : FiniteLaw B) (A : B → Finset X)
{ρ τ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ < 1) (hτ : 0 ≤ τ) :
bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).prob
(fun cb => resamplingZ ν A ρ (resamplingY A W cb.1) < τ)) ≤ τ := by
classical
rw [resampling_density_prob ν A hν hρ (fun Y => resamplingZ ν A ρ Y < τ)]
let μ : FiniteLaw (Finset X) := bernoulliLaw ρ hρ.le hρ1.le
change μ.expectation _ ≤ τ
calc
_ ≤ μ.expectation (fun _ => τ) := by
apply μ.expectation_mono
intro Y
split_ifs with hY
· exact hY.le
· exact hτ
_ = τ := μ.expectation_const τ
theorem local_failure_bound (ν : FiniteLaw B) (A : B → Finset X)
{ρ a : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(m : ℕ) (hm : 1 ≤ m) (hsize : ∀ b, (A b).card ≤ m)
(hspread : IsSpread ν A a) (ha : 0 ≤ a) (hratio : a / ρ ≤ Real.exp (-50)) :
bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).prob
(fun cb => LocalFailure ν A ρ m W cb.1 cb.2)) ≤ Real.exp (-9 * (m : ℝ)) := by
classical
let μ : FiniteLaw (Finset X) := bernoulliLaw ρ hρ.le hρ1.le
let low (W : Finset X) (cb : B × B) : Prop :=
resamplingZ ν A ρ (resamplingY A W cb.1) < Real.exp (-10 * (m : ℝ))
let big (W : Finset X) (cb : B × B) : Prop :=
Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (resamplingY A W cb.1) ∧
m / 2 < (A cb.2 \ W).card
have hpoint (W : Finset X) :
(resamplingLaw ν A ρ W hν hρ).prob (fun cb => LocalFailure ν A ρ m W cb.1 cb.2) ≤
(resamplingLaw ν A ρ W hν hρ).prob (low W) +
(resamplingLaw ν A ρ W hν hρ).prob (big W) := by
apply le_trans ((resamplingLaw ν A ρ W hν hρ).prob_mono (Q := fun cb => low W cb ∨ big W cb) ?_)
((resamplingLaw ν A ρ W hν hρ).prob_union_le (low W) (big W))
intro cb hcb
rcases (localFailure_iff ν A ρ m W cb.1 cb.2).mp hcb with hlow | hfrag
· exact Or.inl hlow
· by_cases hl : low W cb
· exact Or.inl hl
· exact Or.inr ⟨le_of_not_gt hl, hfrag⟩
calc
_ ≤ μ.expectation (fun W => (resamplingLaw ν A ρ W hν hρ).prob (low W) +
(resamplingLaw ν A ρ W hν hρ).prob (big W)) := μ.expectation_mono hpoint
_ = μ.expectation (fun W => (resamplingLaw ν A ρ W hν hρ).prob (low W)) +
μ.expectation (fun W => (resamplingLaw ν A ρ W hν hρ).prob (big W)) :=
μ.expectation_add _ _
_ ≤ Real.exp (-10 * (m : ℝ)) +
Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) :=
add_le_add (resampling_small_density_bound ν A hν hρ hρ1 (Real.exp_pos _).le)
(resampling_large_fragment_bound ν A hν hρ hρ1 m hsize hspread ha hratio)
_ ≤ Real.exp (-9 * (m : ℝ)) := local_failure_exponential_combine hm
theorem local_failure_probability_bound (ν : FiniteLaw B) (A : B → Finset X)
{ρ a : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(m : ℕ) (hm : 1 ≤ m) (hsize : ∀ b, (A b).card ≤ m)
(hspread : IsSpread ν A a) (ha : 0 ≤ a) (hratio : a / ρ ≤ Real.exp (-50)) :
(resamplingExperimentLaw ν A ρ hν hρ hρ1.le).prob
(fun ω => LocalFailure ν A ρ m ω.1 ω.2.1 ω.2.2) ≤ Real.exp (-9 * (m : ℝ)) := by
classical
have heq : (resamplingExperimentLaw ν A ρ hν hρ hρ1.le).prob
(fun ω => LocalFailure ν A ρ m ω.1 ω.2.1 ω.2.2) =
bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).prob
(fun cb => LocalFailure ν A ρ m W cb.1 cb.2)) := by
simp only [FiniteLaw.prob, resamplingExperimentLaw, Fintype.sum_prod_type,
bernoulliExpectation, resamplingLaw_weight, Finset.mul_sum]
apply Finset.sum_congr rfl
intro W _
apply Finset.sum_congr rfl
intro c _
apply Finset.sum_congr rfl
intro b _
by_cases h : LocalFailure ν A ρ m W c b <;> simp [h]
rw [heq]
exact local_failure_bound ν A hν hρ hρ1 m hm hsize hspread ha hratio
omit [Fintype X] in
theorem localFailure_iff_real (ν : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) (W : Finset X) (c b : B) :
LocalFailure ν A ρ m W c b ↔
resamplingZ ν A ρ (resamplingY A W c) < Real.exp (-10 * (m : ℝ)) ∨
(m : ℝ) / 2 < ((A b \ W).card : ℝ) := by
rw [localFailure_iff_twice]
apply or_congr Iff.rfl
constructor
· intro h
have h' : (m : ℝ) < 2 * ((A b \ W).card : ℝ) := by exact_mod_cast h
linarith
· intro h
have h' : (m : ℝ) < 2 * ((A b \ W).card : ℝ) := by linarith
exact_mod_cast h'
theorem local_failure_source_form (ν : FiniteLaw B) (A : B → Finset X)
{ρ a : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(m : ℕ) (hm : 1 ≤ m) (hsize : ∀ b, (A b).card ≤ m)
(hspread : IsSpread ν A a) (ha : 0 ≤ a) (hratio : a / ρ ≤ Real.exp (-50)) :
(resamplingExperimentLaw ν A ρ hν hρ hρ1.le).prob
(fun ω => resamplingZ ν A ρ (resamplingY A ω.1 ω.2.1) <
Real.exp (-10 * (m : ℝ)) ∨
(m : ℝ) / 2 < ((A ω.2.2 \ ω.1).card : ℝ)) ≤ Real.exp (-9 * (m : ℝ)) := by
simpa only [localFailure_iff_real] using
local_failure_probability_bound ν A hν hρ hρ1 m hm hsize hspread ha hratio
end LeanBlast.SecondKahnKalai
end
end OAI
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