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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
def familyKill.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι]
(S : Finset (Assignment u)) (cs : ι → Clause u s) : ℝ :=
if ∃ b ∈ S, ∀ i, Satisfies (cs i) b then 0 else 1
lemma familyKill_nonneg.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι]
(S : Finset (Assignment u)) (cs : ι → Clause u s) : 0 ≤ familyKill S cs := by
unfold familyKill
split <;> norm_num
lemma familyKill_le_one.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι]
(S : Finset (Assignment u)) (cs : ι → Clause u s) : familyKill S cs ≤ 1 := by
unfold familyKill
split <;> norm_num
lemma familyKill_residual.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι]
(S : Finset (Assignment u)) (cs : ι → Clause u s) :
familyKill S cs = 1-alive u (residual S cs) := by
have he : (residual S cs).Nonempty ↔ ∃ b ∈ S, ∀ i, Satisfies (cs i) b := by
simp only [Finset.Nonempty, residual, Finset.mem_filter]
unfold familyKill alive
simp only [he]
split <;> norm_num
lemma familyKill_equiv.{u_1, u_2} {u s : ℕ} {ι : Type u_1} {κ : Type u_2} [Fintype ι] [Fintype κ]
(S : Finset (Assignment u)) (cs : κ → Clause u s) (e : ι ≃ κ) :
familyKill S (fun i => cs (e i)) = familyKill S cs := by
have he : (∃ b ∈ S, ∀ i, Satisfies (cs (e i)) b) ↔ ∃ b ∈ S, ∀ i, Satisfies (cs i) b := by
constructor
· rintro ⟨b,hb,hi⟩
exact ⟨b,hb, fun j => by simpa using hi (e.symm j)⟩
· rintro ⟨b,hb,hi⟩
exact ⟨b,hb, fun j => hi (e j)⟩
simp only [familyKill, he]
lemma familyKill_avg_fin {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) (m : ℕ) :
favg (fun cs : Fin m → Clause u s => familyKill S cs) = blockKill u s S m := by
let := clause_nonempty u s hsu
simp_rw [familyKill_residual]
rw [favg_sub, favg_const, blockKill_iterate hsu, clauseStep_iterate_avg]
lemma familyKill_avg.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι] (hsu : s ≤ u)
(S : Finset (Assignment u)) :
favg (fun cs : ι → Clause u s => familyKill S cs) = blockKill u s S (Fintype.card ι) := by
let e := (Fintype.equivFin ι).symm.arrowCongr (Equiv.refl (Clause u s))
rw [← favg_equiv e]
have he (cs : Fin (Fintype.card ι) → Clause u s) :
familyKill S (e cs) = familyKill S cs :=
familyKill_equiv S cs (Fintype.equivFin ι)
simp_rw [he]
exact familyKill_avg_fin hsu S _
lemma blockKill_mono_length {u s : ℕ} (hsu : s ≤ u) (S : Finset (Assignment u)) :
Monotone (blockKill u s S) := by
let := streamLaw_probability u s hsu
intro m d hmd
unfold blockKill
apply ENNReal.toReal_mono (measure_ne_top _ _)
apply measure_mono
intro ω hm hd
rcases hd with ⟨b,hb,ht⟩
exact hm ⟨b,hb,fun i hi => ht i (lt_of_lt_of_le hi hmd)⟩
lemma restricted_familyKill_avg.{u_1} {u s : ℕ} {ι : Type u_1} [Fintype ι] [DecidableEq ι]
(hsu : s ≤ u) (S : Finset (Assignment u)) (A : Finset ι) :
favg (fun cs : ι → Clause u s => familyKill S (fun i : A => cs i)) =
blockKill u s S A.card := by
let := clause_nonempty u s hsu
rw [favg_pi_restrict (fun i => i ∈ A) (familyKill S)]
convert (familyKill_avg (ι := A) hsu S) using 1
· congr 1
exact Subsingleton.elim _ _
· rw [Fintype.card_coe]
def testIndices {u k M : ℕ} (a : Assignment k) (env : Fin M → RootData u k)
(m : ℕ) (j : Fin k) : Finset (Fin M) :=
Finset.univ.filter fun i => i.val < m ∧ TestAt a (env i) j
lemma forcedPrefix_familyKill {u k M : ℕ} (a : Assignment k) (env : Fin M → RootData u k)
(o : Fin M → Clause u k) (ys : Fin k → Fin M → Clause u (k-1)) (m : ℕ)
(hforced : ∀ x ∈ pairPrefix Finset.univ
(fun i => refreshData a (env i) (o i, fun j => ys j i)) m, x.1 = a)
(j : Fin k) (hj : Available a env j) :
familyKill (fibrePrefix a env o m) (fun i : testIndices a env m j => ys j i) = 1 := by
unfold familyKill
rw [ite_eq_right]
rintro ⟨b,hb,ht⟩
apply forcedPrefix_testKills a env o ys m hforced j hj
refine ⟨b,hb,fun i hi hit => ?_⟩
exact ht ⟨i, by simp [testIndices, hi, hit]⟩
lemma prod_filter_indicator.{u_1} {ι : Type u_1} [Fintype ι] [DecidableEq ι] (A : Finset ι) (q : ℝ) :
(∏ j : ι, if j ∈ A then q else 1) = q^A.card := by
classical
rw [← Finset.prod_filter]
simp
lemma forcedPrefix_probability {u k M d : ℕ} (hku : k ≤ u) (a : Assignment k)
(env : Fin M → RootData u k) (o : Fin M → Clause u k) (m : ℕ)
(J : Finset (Fin k)) (hJ : ∀ j ∈ J, Available a env j)
(hd : ∀ j, (testIndices a env m j).card ≤ d) :
favg (fun ys : Fin k → Fin M → Clause u (k-1) =>
forcedReward a (pairPrefix Finset.univ
(fun i => refreshData a (env i) (o i, fun j => ys j i)) m)) ≤
alive u (fibrePrefix a env o m) * blockKill u (k-1) (fibrePrefix a env o m) d ^ J.card := by
let := clause_nonempty u (k-1) (by omega)
let S := fibrePrefix a env o m
let q := blockKill u (k-1) S d
let F (j : Fin k) (w : Fin M → Clause u (k-1)) : ℝ :=
if j ∈ J then familyKill S (fun i : testIndices a env m j => w i) else 1
have hF : ∀ j w, 0 ≤ F j w := by
intro j w
dsimp only [F]
split
· exact familyKill_nonneg _ _
· norm_num
have hp (ys : Fin k → Fin M → Clause u (k-1)) :
forcedReward a (pairPrefix Finset.univ
(fun i => refreshData a (env i) (o i, fun j => ys j i)) m) ≤
alive u S * ∏ j, F j (ys j) := by
unfold forcedReward
split_ifs with h
· have hprod : (∏ j, F j (ys j)) = 1 := by
apply Finset.prod_eq_one
intro j _
dsimp only [F]
split_ifs with hj
· exact forcedPrefix_familyKill a env o ys m h.2 j (hJ j hj)
· rfl
rw [hprod, mul_one]
have hh := forcedReward_le_alive a (pairPrefix Finset.univ
(fun i => refreshData a (env i) (o i, fun j => ys j i)) m)
rw [fibre_pairPrefix_refresh] at hh
simpa only [forcedReward, ite_eq_left h] using hh
· exact mul_nonneg (alive_nonneg _ _) (Finset.prod_nonneg fun j _ => hF j _)
have havg (j : Fin k) : favg (F j) ≤ if j ∈ J then q else 1 := by
by_cases hj : j ∈ J
· simp only [F, ite_eq_left hj]
rw [restricted_familyKill_avg (by omega)]
exact blockKill_mono_length (by omega) S (hd j)
· simp only [F, ite_eq_right hj, favg_const, le_refl]
calc
_ ≤ favg (fun ys : Fin k → Fin M → Clause u (k-1) => alive u S * ∏ j, F j (ys j)) :=
favg_mono hp
_ = alive u S * ∏ j, favg (F j) := by rw [favg_mul, favg_pi_product]
_ ≤ alive u S * ∏ j : Fin k, if j ∈ J then q else 1 := by
apply mul_le_mul_of_nonneg_left _ (alive_nonneg _ _)
exact Finset.prod_le_prod₀ (fun j _ => favg_nonneg (hF j)) (fun j _ => havg j)
_ = _ := by rw [prod_filter_indicator]
end RandomKSAT
end
end OAI
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