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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace RandomKSAT
open scoped Classical ENNReal
def extendWord.{u_1} {α : Type u_1} [Nonempty α] {M : ℕ} (w : Fin M → α) (i : ℕ) : α :=
if h : i < M then w ⟨i,h⟩ else Classical.choice ‹Nonempty α›
lemma extendWord_fin.{u_1} {α : Type u_1} [Nonempty α] {M : ℕ} (w : Fin M → α) (i : Fin M) :
extendWord w i = w i := by simp [extendWord, i.isLt]
def wordCount.{u_1} {α : Type u_1} {M : ℕ} (p : α → Prop) (w : Fin M → α) : ℝ :=
∑ i, if p (w i) then 1 else 0
def hitProbability.{u_1} {α : Type u_1} [Fintype α] (p : α → Prop) : ℝ :=
favg (fun x => if p x then 1 else 0)
lemma wordCount_nonneg.{u_1} {α : Type u_1} {M : ℕ} (p : α → Prop) (w : Fin M → α) :
0 ≤ wordCount p w := Finset.sum_nonneg fun _ _ => by split <;> norm_num
lemma hitProbability_nonneg.{u_1} {α : Type u_1} [Fintype α] (p : α → Prop) :
0 ≤ hitProbability p := favg_nonneg fun _ => by split <;> norm_num
lemma wordCount_cons.{u_1} {α : Type u_1} {M : ℕ} (p : α → Prop) (x : α) (w : Fin M → α) :
wordCount p (Fin.cons x w) = (if p x then 1 else 0)+wordCount p w := by
rw [wordCount, Fin.sum_univ_succ]
simp only [Fin.cons_zero, Fin.cons_succ]
rfl
lemma wordCount_zero.{u_1} {α : Type u_1} (p : α → Prop) (w : Fin 0 → α) : wordCount p w = 0 := by
simp [wordCount]
lemma exp_hit_avg.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p : α → Prop) :
favg (fun x => Real.exp (if p x then 1 else 0)) =
1+(Real.exp 1-1)*hitProbability p := by
have he (x : α) : Real.exp (if p x then 1 else 0) =
1+(Real.exp 1-1)*(if p x then 1 else 0) := by split <;> simp
simp only [he, favg_add, favg_const, favg_mul, hitProbability]
lemma subhit_exp_avg.{u_1} {α : Type u_1} [Fintype α] (p q : α → Prop) (hqp : ∀ x, q x → p x) :
favg (fun x => (if q x then 1 else 0)*Real.exp (if p x then 1 else 0)) =
Real.exp 1*hitProbability q := by
have he (x : α) : (if q x then 1 else 0)*Real.exp (if p x then 1 else 0) =
Real.exp 1*(if q x then 1 else 0) := by
by_cases hq : q x
· simp [hq, hqp x hq]
· simp [hq]
simp only [he, favg_mul, hitProbability]
lemma exp_wordCount_avg.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p : α → Prop) (M : ℕ) :
favg (fun w : Fin M → α => Real.exp (wordCount p w)) =
(1+(Real.exp 1-1)*hitProbability p)^M := by
induction M with
| zero => simp only [wordCount_zero, Real.exp_zero, favg_const, pow_zero]
| succ M ih =>
rw [favg_cons, favg_comm]
simp_rw [wordCount_cons, Real.exp_add, favg_mul, ih]
have he (x : α) : Real.exp (if p x then 1 else 0) *
(1+(Real.exp 1-1)*hitProbability p)^M =
(1+(Real.exp 1-1)*hitProbability p)^M * Real.exp (if p x then 1 else 0) := mul_comm _ _
simp only [he, favg_mul, exp_hit_avg, pow_succ]
lemma exp_wordCount_bound.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p : α → Prop) (M : ℕ) :
favg (fun w : Fin M → α => Real.exp (wordCount p w)) ≤
Real.exp (M*(Real.exp 1-1)*hitProbability p) := by
rw [exp_wordCount_avg]
have he : 1+(Real.exp 1-1)*hitProbability p ≤
Real.exp ((Real.exp 1-1)*hitProbability p) := by
simpa only [add_comm] using Real.add_one_le_exp ((Real.exp 1-1)*hitProbability p)
have hp : 0 ≤ 1+(Real.exp 1-1)*hitProbability p := by
have he1 := Real.exp_one_gt_d9
nlinarith [hitProbability_nonneg p]
have hh := pow_le_pow_left₀ hp he M
rw [← Real.exp_nat_mul] at hh
simpa only [mul_assoc] using hh
lemma wordCount_avg.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p : α → Prop) (M : ℕ) :
favg (fun w : Fin M → α => wordCount p w) = M*hitProbability p := by
induction M with
| zero => simp only [wordCount_zero, favg_const, Nat.cast_zero, zero_mul]
| succ M ih =>
rw [favg_cons]
simp only [wordCount_cons, favg_add, favg_const]
change hitProbability p+favg (fun w : Fin M → α => wordCount p w) = _
rw [ih, Nat.cast_add, Nat.cast_one]
ring
lemma subcount_exp_bound.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p q : α → Prop)
(hqp : ∀ x, q x → p x) (M : ℕ) :
favg (fun w : Fin M → α => wordCount q w * Real.exp (wordCount p w)) ≤
M*Real.exp 1*hitProbability q*(1+(Real.exp 1-1)*hitProbability p)^M := by
let A := 1+(Real.exp 1-1)*hitProbability p
have hA : 1 ≤ A := by
dsimp only [A]
nlinarith [Real.exp_one_gt_d9, hitProbability_nonneg p]
have hq := hitProbability_nonneg q
induction M with
| zero => simp only [wordCount_zero, zero_mul, favg_const, Nat.cast_zero, le_refl]
| succ M ih =>
rw [favg_cons]
have hex (w : Fin M → α) (x : α) :
wordCount q (Fin.cons x w)*Real.exp (wordCount p (Fin.cons x w)) =
Real.exp (wordCount p w)*((if q x then 1 else 0)*Real.exp (if p x then 1 else 0)) +
(wordCount q w*Real.exp (wordCount p w))*Real.exp (if p x then 1 else 0) := by
rw [wordCount_cons, wordCount_cons, Real.exp_add]
ring
simp only [hex, favg_add, favg_mul, subhit_exp_avg p q hqp, exp_hit_avg]
have he (w : Fin M → α) :
Real.exp (wordCount p w)*(Real.exp 1*hitProbability q) =
(Real.exp 1*hitProbability q)*Real.exp (wordCount p w) := mul_comm _ _
have he' (w : Fin M → α) :
wordCount q w*Real.exp (wordCount p w)*(1+(Real.exp 1-1)*hitProbability p) =
A*(wordCount q w*Real.exp (wordCount p w)) := mul_comm _ _
simp only [he, he', favg_mul, exp_wordCount_avg]
change Real.exp 1*hitProbability q*A^M + A*favg (fun w : Fin M → α => wordCount q w*Real.exp (wordCount p w)) ≤ _
have hmul := mul_le_mul_of_nonneg_left ih (by linarith : 0 ≤ A)
have hpow : 0 ≤ Real.exp 1*hitProbability q*A^M := by positivity
have hextra := mul_le_mul_of_nonneg_right hA hpow
rw [Nat.cast_add, Nat.cast_one, pow_succ]
change _ ≤ (↑M+1)*Real.exp 1*hitProbability q*(A^M*A)
nlinarith
lemma wordCount_factorial_second.{u_1} {α : Type u_1} [Fintype α] [Nonempty α]
(p : α → Prop) (M : ℕ) :
favg (fun w : Fin M → α => wordCount p w*(wordCount p w-1)) =
(M : ℝ)*(M-1)*(hitProbability p)^2 := by
induction M with
| zero => simp only [wordCount_zero, zero_mul, favg_const, Nat.cast_zero]
| succ M ih =>
rw [favg_cons]
have he (w : Fin M → α) (x : α) :
wordCount p (Fin.cons x w)*(wordCount p (Fin.cons x w)-1) =
wordCount p w*(wordCount p w-1)+(2*wordCount p w)*(if p x then 1 else 0) := by
rw [wordCount_cons]
split <;> ring
simp only [he, favg_add, favg_const, favg_mul]
change favg (fun w : Fin M → α => wordCount p w*(wordCount p w-1)) +
favg (fun w : Fin M → α => (2*wordCount p w)*hitProbability p) = _
have he' (w : Fin M → α) : (2*wordCount p w)*hitProbability p =
(2*hitProbability p)*wordCount p w := by ring
simp only [he', favg_mul, wordCount_avg, ih, Nat.cast_add, Nat.cast_one]
ring
lemma one_add_cube_le_exp {x : ℝ} (hx : 0 ≤ x) :
(1+x)^3 ≤ 6*Real.exp 1*Real.exp x := by
have hh := Real.pow_div_factorial_le_exp (1+x) (by linarith) 3
norm_num only [Nat.factorial, Nat.cast_ofNat] at hh
rw [Real.exp_add] at hh
linarith
lemma one_add_sq_le_exp {x : ℝ} (hx : 0 ≤ x) :
(1+x)^2 ≤ 2*Real.exp 1*Real.exp x := by
have hh := Real.pow_div_factorial_le_exp (1+x) (by linarith) 2
norm_num only [Nat.factorial, Nat.cast_ofNat] at hh
rw [Real.exp_add] at hh
linarith
lemma wordCount_cube_bound.{u_1} {α : Type u_1} [Fintype α] [Nonempty α]
(p : α → Prop) (M : ℕ) {A : ℝ} (hA : M*hitProbability p ≤ A) :
favg (fun w : Fin M → α => (1+wordCount p w)^3) ≤
6*Real.exp 1*Real.exp ((Real.exp 1-1)*A) := by
have hh := favg_mono (fun w : Fin M → α => one_add_cube_le_exp (wordCount_nonneg p w))
rw [favg_mul] at hh
have he := exp_wordCount_bound p M
have ha : Real.exp (M*(Real.exp 1-1)*hitProbability p) ≤ Real.exp ((Real.exp 1-1)*A) := by
apply Real.exp_le_exp.mpr
nlinarith [Real.exp_one_gt_d9]
exact hh.trans (mul_le_mul_of_nonneg_left (he.trans ha) (by positivity))
lemma subcount_sq_bound.{u_1} {α : Type u_1} [Fintype α] [Nonempty α] (p q : α → Prop)
(hqp : ∀ x, q x → p x) (M : ℕ) {A : ℝ} (hA : M*hitProbability p ≤ A) :
favg (fun w : Fin M → α => wordCount q w*(1+wordCount p w)^2) ≤
2*(Real.exp 1)^2*M*hitProbability q*Real.exp ((Real.exp 1-1)*A) := by
have hh := favg_mono (fun w : Fin M → α =>
mul_le_mul_of_nonneg_left (one_add_sq_le_exp (wordCount_nonneg p w)) (wordCount_nonneg q w))
have hew (w : Fin M → α) : wordCount q w*(2*Real.exp 1*Real.exp (wordCount p w)) =
(2*Real.exp 1)*(wordCount q w*Real.exp (wordCount p w)) := by ring
simp only [hew, favg_mul] at hh
have hs := subcount_exp_bound p q hqp M
rw [← exp_wordCount_avg p M] at hs
have he := exp_wordCount_bound p M
have ha : Real.exp (M*(Real.exp 1-1)*hitProbability p) ≤ Real.exp ((Real.exp 1-1)*A) := by
apply Real.exp_le_exp.mpr
nlinarith [Real.exp_one_gt_d9]
have hq := hitProbability_nonneg q
have hf := mul_le_mul_of_nonneg_left (he.trans ha)
(show 0 ≤ M*Real.exp 1*hitProbability q by positivity)
have hg := mul_le_mul_of_nonneg_left (hs.trans hf) (show 0 ≤ 2*Real.exp 1 by positivity)
nlinarith
lemma wordCount_factorial_second_bound.{u_1} {α : Type u_1} [Fintype α] [Nonempty α]
(p : α → Prop) (M : ℕ) :
favg (fun w : Fin M → α => wordCount p w*(wordCount p w-1)) ≤
(M : ℝ)^2*(hitProbability p)^2 := by
rw [wordCount_factorial_second]
nlinarith [sq_nonneg (hitProbability p), Nat.cast_nonneg (α := ℝ) M]
def rootIncident {u k : ℕ} (p : RootData u k) : Prop := rootSize p.1 ≠ 0
def rootCollision {u k : ℕ} (p : RootData u k) : Prop := 2 ≤ rootSize p.1
lemma wordCount_incident {u k M : ℕ} (ps : Fin M → RootData u k) :
wordCount rootIncident ps = (incidentSet ps).card := by
simp only [wordCount, rootIncident, incidentSet]
norm_cast
exact Finset.sum_boole _ _
lemma wordCount_collision {u k M : ℕ} (ps : Fin M → RootData u k) :
wordCount rootCollision ps = (collisionSet ps).card := by
simp only [wordCount, rootCollision, collisionSet]
norm_cast
exact Finset.sum_boole _ _
lemma rootRewards_square_poly {u r M : ℕ} (hr : 2 ≤ r) (hru : r+1 ≤ u)
(a : Assignment (r+1)) (ps : Fin M → RootData u (r+1)) :
favg (fun z : Fin M → RefreshNoise u (r+1) =>
(dataRewards (forcedReward a) Finset.univ (fun i => refreshData a (ps i) (z i)))^2) ≤
8*(lifetimeC (r+1)+1)^2*durationLog (r+1) M*(1+wordCount rootIncident ps)^3 +
2*M*(lifetimeC (r+1)+1)*(wordCount rootCollision ps*(1+wordCount rootIncident ps)^2) +
(M : ℝ)^2*(wordCount rootCollision ps*(wordCount rootCollision ps-1)) := by
let hne (t : RootType (r+1)) : Nonempty (Clause u ((r+1)-rootSize t)) := clause_nonempty _ _ (by omega)
let : Nonempty (RootData u (r+1)) := ⟨⟨fun _ => none, Classical.choice (hne _)⟩⟩
have hh := rootRewards_square_conditioned (M := M) hr hru a (extendWord ps)
simp only [extendWord_fin] at hh
let Z := (incidentSet ps).card
let V : ℝ := (collisionSet ps).card
let Q := deletionQ (r+1) (max 1 Z)
let C := lifetimeC (r+1)+1
have hV : 0 ≤ V := Nat.cast_nonneg _
have hlog : 0 ≤ durationLog (r+1) M := le_trans (by norm_num) (durationLog_one_le _ _)
have hq := deletionQ_poly_bound (Z := Z) hr
have hq2 := deletionQ_sq_poly_bound (Z := Z) hr
change Q ≤ C*(1+(Z : ℝ))^2 at hq
change Q^2 ≤ C^2*(1+(Z : ℝ))^3 at hq2
have hfirst := mul_le_mul_of_nonneg_left hq2 (show 0 ≤ 8*durationLog (r+1) M by positivity)
have hsecond := mul_le_mul_of_nonneg_left hq (show 0 ≤ 2*M*V by positivity)
rw [wordCount_incident, wordCount_collision]
change _ ≤ 8*C^2*durationLog (r+1) M*(1+(Z : ℝ))^3 +
2*M*C*(V*(1+(Z : ℝ))^2)+(M : ℝ)^2*(V*(V-1))
change _ ≤ 8*Q^2*durationLog (r+1) M+2*M*Q*V+(M : ℝ)^2*V*(V-1) at hh
nlinarith
def deletionMomentC (k : ℕ) (A D : ℝ) : ℝ :=
48*Real.exp 1*(lifetimeC k+1)^2*Real.exp ((Real.exp 1-1)*A) +
4*(Real.exp 1)^2*(lifetimeC k+1)*D*Real.exp ((Real.exp 1-1)*A)+D^2
lemma deletionMomentC_nonneg {r : ℕ} (hr : 2 ≤ r) (A D : ℝ) (hD : 0 ≤ D) :
0 ≤ deletionMomentC (r+1) A D := by
have hc : 0 ≤ lifetimeC (r+1)+1 := by linarith [lifetimeC_pos (by omega : 1 ≤ r)]
unfold deletionMomentC
positivity
lemma rootRewards_square_avg {u r M : ℕ} (hr : 2 ≤ r) (hru : r+1 ≤ u)
(a : Assignment (r+1)) {A D : ℝ}
(hA : M*hitProbability (rootIncident (u := u) (k := r+1)) ≤ A)
(hD : (M : ℝ)^2*hitProbability (rootCollision (u := u) (k := r+1)) ≤ D) :
favg (fun ps : Fin M → RootData u (r+1) => (dataRewards (forcedReward a) Finset.univ ps)^2) ≤
deletionMomentC (r+1) A D*durationLog (r+1) M := by
let hne (t : RootType (r+1)) : Nonempty (Clause u ((r+1)-rootSize t)) := clause_nonempty _ _ (by omega)
let : Nonempty (RootData u (r+1)) := ⟨⟨fun _ => none, Classical.choice (hne _)⟩⟩
let C := lifetimeC (r+1)+1
let h := durationLog (r+1) M
let E := Real.exp ((Real.exp 1-1)*A)
let q := hitProbability (rootCollision (u := u) (k := r+1))
have hC : 0 ≤ C := by dsimp [C]; linarith [lifetimeC_pos (by omega : 1 ≤ r)]
have hE : 0 ≤ E := (Real.exp_pos _).le
have hq : 0 ≤ q := hitProbability_nonneg _
have hh : 1 ≤ h := durationLog_one_le _ _
have hD0 : 0 ≤ D := le_trans (mul_nonneg (sq_nonneg (M : ℝ)) hq) hD
have hstat := favg_pointwise_refresh (refreshData a) (favg_refreshData hru a) M
(fun ps : Fin M → RootData u (r+1) => (dataRewards (forcedReward a) Finset.univ ps)^2)
have hpoly := favg_mono (fun ps : Fin M → RootData u (r+1) => rootRewards_square_poly hr hru a ps)
rw [hstat] at hpoly
simp only [favg_add, favg_mul] at hpoly
have h3 := wordCount_cube_bound (rootIncident (u := u) (k := r+1)) M hA
have h2 := subcount_sq_bound (rootIncident (u := u) (k := r+1)) rootCollision
(fun _ hc => by unfold rootIncident rootCollision at *; omega) M hA
have hf := wordCount_factorial_second_bound (rootCollision (u := u) (k := r+1)) M
have h3' := mul_le_mul_of_nonneg_left h3 (show 0 ≤ 8*C^2*h by positivity)
have h2' := mul_le_mul_of_nonneg_left h2 (show 0 ≤ 2*M*C by positivity)
have hf' := mul_le_mul_of_nonneg_left hf (sq_nonneg (M : ℝ))
have hD' := mul_le_mul_of_nonneg_left hD (show 0 ≤ 4*(Real.exp 1)^2*C*E by positivity)
have hDsq := pow_le_pow_left₀ (show 0 ≤ (M : ℝ)^2*q by positivity) hD 2
have hstretch := mul_le_mul_of_nonneg_right hh
(show 0 ≤ 4*(Real.exp 1)^2*C*D*E+D^2 by positivity)
change _ ≤ (48*Real.exp 1*C^2*E+4*(Real.exp 1)^2*C*D*E+D^2)*h
change _ ≤ 8*C^2*h*favg (fun ps : Fin M → RootData u (r+1) => (1+wordCount rootIncident ps)^3) +
2*M*C*favg (fun ps : Fin M → RootData u (r+1) => wordCount rootCollision ps*(1+wordCount rootIncident ps)^2)+
(M : ℝ)^2*favg (fun ps : Fin M → RootData u (r+1) => wordCount rootCollision ps*(wordCount rootCollision ps-1)) at hpoly
dsimp only [E, q] at hDsq hD' hstretch
dsimp only [C, h, E] at hpoly ⊢
nlinarith
end RandomKSAT
end
end OAI
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