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760f90e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 | import OAI.Computability.StarHeight.Actions
namespace OAI
namespace GeneralizedStarHeight
universe u
universe uΩ uι
namespace FiniteAvoidance
variable {Ω : Type uΩ} {ι : Type uι} [Fintype Ω]
noncomputable def mass (w : Ω → ℝ) (A : Set Ω) : ℝ := by
classical
exact ∑ x : Ω, if x ∈ A then w x else 0
lemma mass_nonneg {w : Ω → ℝ} (hw : ∀ x, 0 ≤ w x) (A : Set Ω) :
0 ≤ mass w A := by
classical
exact Finset.sum_nonneg fun x _ => by
by_cases hx : x ∈ A <;> simp [hx, hw x]
lemma mass_mono {w : Ω → ℝ} (hw : ∀ x, 0 ≤ w x) {A C : Set Ω}
(hAC : A ⊆ C) : mass w A ≤ mass w C := by
classical
apply Finset.sum_le_sum
intro x _
by_cases hx : x ∈ A
· simp [hx, hAC hx]
· by_cases hc : x ∈ C <;> simp [hx, hc, hw x]
@[simp] lemma mass_empty (w : Ω → ℝ) : mass w ∅ = 0 := by
classical
simp [mass]
lemma mass_diff (w : Ω → ℝ) (A C : Set Ω) :
mass w (A \ C) = mass w A - mass w (C ∩ A) := by
classical
simp only [mass, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro x _
by_cases hx : x ∈ A <;> by_cases hc : x ∈ C <;> simp [hx, hc]
def avoids (B : ι → Set Ω) (J : Finset ι) : Set Ω :=
{x | ∀ i ∈ J, x ∉ B i}
omit [Fintype Ω] in
@[simp] lemma avoids_empty (B : ι → Set Ω) : avoids B ∅ = Set.univ := by
ext x
simp [avoids]
omit [Fintype Ω] in
lemma avoids_antitone (B : ι → Set Ω) {J K : Finset ι} (hJK : J ⊆ K) :
avoids B K ⊆ avoids B J := by
intro x hx i hi
exact hx i (hJK hi)
omit [Fintype Ω] in
lemma avoids_insert [DecidableEq ι] (B : ι → Set Ω) (J : Finset ι) (i : ι) :
avoids B (insert i J) = avoids B J \ B i := by
ext x
simp only [avoids, Set.mem_ofPred_eq, Finset.mem_insert, forall_eq_or_imp,
Set.mem_sdiff]
exact and_comm
lemma mass_avoids_insert [DecidableEq ι] (w : Ω → ℝ) (B : ι → Set Ω)
(J : Finset ι) (i : ι) :
mass w (avoids B (insert i J)) =
mass w (avoids B J) - mass w (B i ∩ avoids B J) := by
rw [avoids_insert, mass_diff]
lemma avoid_union_lower [DecidableEq ι] (w : Ω → ℝ) (B : ι → Set Ω)
(s : ℝ) (hs : s ≤ 1) (J K : Finset ι) (hd : Disjoint J K)
(hb : ∀ U : Finset ι, U ⊆ K → U.card < K.card →
∀ i, i ∉ J ∪ U → mass w (B i ∩ avoids B (J ∪ U)) ≤
s * mass w (avoids B (J ∪ U))) :
(1 - s) ^ K.card * mass w (avoids B J) ≤ mass w (avoids B (J ∪ K)) := by
induction K using Finset.induction_on with
| empty => simp
| @insert i K hi ih =>
have hdK : Disjoint J K := hd.mono_right (Finset.subset_insert _ _)
have hiJ : i ∉ J := by
intro hm
exact Finset.disjoint_left.mp hd hm (Finset.mem_insert_self _ _)
have hbK : ∀ U : Finset ι, U ⊆ K → U.card < K.card →
∀ j, j ∉ J ∪ U → mass w (B j ∩ avoids B (J ∪ U)) ≤
s * mass w (avoids B (J ∪ U)) := by
intro U hU hcard j hj
apply hb U (hU.trans (Finset.subset_insert _ _))
· simpa [Finset.card_insert_of_notMem hi] using hcard.trans_le (Nat.le_succ _)
· exact hj
have hstep := hb K (Finset.subset_insert _ _) (by simp [hi]) i
(by simp [hiJ, hi])
have hrec := ih hdK hbK
rw [Finset.card_insert_of_notMem hi, pow_succ,
mul_comm ((1 - s) ^ K.card) (1 - s), mul_assoc]
calc
(1 - s) * ((1 - s) ^ K.card * mass w (avoids B J))
≤ (1 - s) * mass w (avoids B (J ∪ K)) :=
mul_le_mul_of_nonneg_left hrec (sub_nonneg.mpr hs)
_ ≤ mass w (avoids B (J ∪ insert i K)) := by
rw [Finset.union_insert, mass_avoids_insert]
linarith
theorem local_lemma [DecidableEq ι] (w : Ω → ℝ) (hw : ∀ x, 0 ≤ w x)
(htotal : 0 < mass w Set.univ) (B : ι → Set Ω)
(neighbors : ι → Finset ι) (D : ℕ) (s : ℝ) (hs0 : 0 < s) (hs1 : s < 1)
(hdegree : ∀ i, (neighbors i).card ≤ D)
(hbad : ∀ i, mass w (B i) ≤ s * (1 - s) ^ D * mass w Set.univ)
(hind : ∀ i (J : Finset ι), i ∉ J → Disjoint J (neighbors i) →
mass w (B i ∩ avoids B J) * mass w Set.univ =
mass w (B i) * mass w (avoids B J))
(J : Finset ι) : 0 < mass w (avoids B J) := by
have hconditional : ∀ n : ℕ, ∀ K : Finset ι, K.card = n →
0 < mass w (avoids B K) ∧ ∀ i, i ∉ K →
mass w (B i ∩ avoids B K) ≤ s * mass w (avoids B K) := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro K hK
have hpos : 0 < mass w (avoids B K) := by
by_cases he : K = ∅
· simpa [he] using htotal
· obtain ⟨i, hi⟩ := Finset.nonempty_iff_ne_empty.mpr he
have hsmall : (K.erase i).card < n := by
rw [← hK]
exact Finset.card_erase_lt_of_mem hi
have hprev := ih (K.erase i).card hsmall (K.erase i) rfl
have hb := hprev.2 i (Finset.notMem_erase i K)
have hins : insert i (K.erase i) = K := Finset.insert_erase hi
rw [← hins, mass_avoids_insert]
have hp : 0 < (1 - s) * mass w (avoids B (K.erase i)) :=
mul_pos (sub_pos.mpr hs1) hprev.1
linarith
refine ⟨hpos, ?_⟩
intro i hi
let N := K ∩ neighbors i
let F := K \ neighbors i
have hFN : Disjoint F N := by
exact Finset.disjoint_sdiff_inter _ _
have hUF : F ∪ N = K := by
exact Finset.sdiff_union_inter _ _
have hFsub : F ⊆ K := Finset.sdiff_subset
have hFdisj : Disjoint F (neighbors i) := by
apply Finset.disjoint_left.mpr
intro x hx hnx
exact (Finset.mem_sdiff.mp hx).2 hnx
have hiF : i ∉ F := fun h => hi (hFsub h)
have hInd := hind i F hiF hFdisj
have hBF : mass w (B i ∩ avoids B F) ≤
s * (1 - s) ^ D * mass w (avoids B F) := by
apply (mul_le_mul_iff_left₀ htotal).mp
rw [hInd]
calc
mass w (B i) * mass w (avoids B F)
≤ (s * (1 - s) ^ D * mass w Set.univ) * mass w (avoids B F) :=
mul_le_mul_of_nonneg_right (hbad i) (mass_nonneg hw _)
_ = (s * (1 - s) ^ D * mass w (avoids B F)) * mass w Set.univ := by ring
have hchain : (1 - s) ^ N.card * mass w (avoids B F) ≤
mass w (avoids B K) := by
rw [← hUF]
apply avoid_union_lower w B s hs1.le F N hFN
intro U hU hcard j hj
have hdu : Disjoint F U := hFN.mono_right hU
have hsmall : (F ∪ U).card < n := by
rw [Finset.card_union_of_disjoint hdu, ← hK, ← hUF,
Finset.card_union_of_disjoint hFN]
exact Nat.add_lt_add_left hcard _
exact (ih (F ∪ U).card hsmall (F ∪ U) rfl).2 j hj
have hNcard : N.card ≤ D :=
(Finset.card_le_card (Finset.inter_subset_right)).trans (hdegree i)
have hpow : (1 - s) ^ D ≤ (1 - s) ^ N.card :=
pow_le_pow_of_le_one (sub_nonneg.mpr hs1.le)
(by linarith) hNcard
calc
mass w (B i ∩ avoids B K) ≤ mass w (B i ∩ avoids B F) :=
mass_mono hw (Set.inter_subset_inter_right _ (avoids_antitone B hFsub))
_ ≤ s * (1 - s) ^ D * mass w (avoids B F) := hBF
_ ≤ s * ((1 - s) ^ N.card * mass w (avoids B F)) := by
have hh := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hpow hs0.le) (mass_nonneg hw (avoids B F))
simpa only [mul_assoc] using hh
_ ≤ s * mass w (avoids B K) := mul_le_mul_of_nonneg_left hchain hs0.le
exact (hconditional J.card J rfl).1
lemma mass_one_eq_card (A : Set Ω) :
mass (fun _ : Ω => (1 : ℝ)) A = Nat.card A := by
classical
simp [mass, Nat.card_eq_fintype_card, Fintype.card_subtype]
@[simp] lemma mass_one_univ :
mass (fun _ : Ω => (1 : ℝ)) Set.univ = Fintype.card Ω := by
classical
simp [mass]
noncomputable def probability (E : Set Ω) : ℝ :=
mass (fun _ => (1 : ℝ)) E / Fintype.card Ω
@[simp] lemma probability_univ [Nonempty Ω] :
probability (Set.univ : Set Ω) = 1 := by
simp [probability, ne_of_gt Fintype.card_pos]
lemma probability_compl [Nonempty Ω] (E : Set Ω) :
probability Eᶜ = 1 - probability E := by
have hd := mass_diff (fun _ : Ω => (1 : ℝ)) Set.univ E
simp only [← Set.compl_eq_univ_sdiff, Set.inter_univ, mass_one_univ] at hd
unfold probability
rw [hd, sub_div, div_self (by exact_mod_cast ne_of_gt Fintype.card_pos)]
lemma probability_eq_of_equiv (E F : Set Ω) (e : E ≃ F) :
probability E = probability F := by
unfold probability
rw [mass_one_eq_card, mass_one_eq_card, Nat.card_congr e]
lemma probability_inter_of_independent [Nonempty Ω] (E F : Set Ω)
(hind : mass (fun _ : Ω => (1 : ℝ)) (E ∩ F) *
mass (fun _ : Ω => (1 : ℝ)) Set.univ =
mass (fun _ : Ω => (1 : ℝ)) E * mass (fun _ : Ω => (1 : ℝ)) F) :
probability (E ∩ F) = probability E * probability F := by
have hp : (Fintype.card Ω : ℝ) ≠ 0 := by
exact_mod_cast ne_of_gt Fintype.card_pos
simp only [mass_one_univ] at hind
unfold probability
field_simp
exact hind
lemma nonempty_of_mass_pos {w : Ω → ℝ} {E : Set Ω} (h : 0 < mass w E) :
E.Nonempty := by
by_contra hn
have he := Set.not_nonempty_iff_eq_empty.mp hn
simp [he] at h
lemma local_parameter (D : ℕ) (p : ℝ)
(hp : 4 * ((D : ℝ) + 1) * p ≤ 1) :
∃ s : ℝ, 0 < s ∧ s < 1 ∧ p ≤ s * (1 - s) ^ D := by
let s : ℝ := (2 * ((D : ℝ) + 1))⁻¹
have hD : (0 : ℝ) ≤ D := Nat.cast_nonneg D
have hden : 0 < 2 * ((D : ℝ) + 1) := by positivity
have hs0 : 0 < s := inv_pos.mpr hden
have heq : s * (2 * ((D : ℝ) + 1)) = 1 := inv_mul_cancel₀ hden.ne'
have hsD : 0 ≤ s * D := mul_nonneg hs0.le hD
have hshalf : s ≤ 1 / 2 := by nlinarith
have hs1 : s < 1 := by linarith
have hbern : 1 - (D : ℝ) * s ≤ (1 - s) ^ D := by
simpa only [sub_eq_add_neg, mul_neg] using
(one_add_mul_le_pow (a := -s) (by linarith) D)
have hhalf : (1 : ℝ) / 2 ≤ (1 - s) ^ D := by nlinarith
have hp' : p ≤ s / 2 := by
apply (mul_le_mul_iff_right₀ (show 0 < 4 * ((D : ℝ) + 1) by positivity)).mp
nlinarith [heq]
refine ⟨s, hs0, hs1, hp'.trans ?_⟩
nlinarith
end FiniteAvoidance
end GeneralizedStarHeight
end OAI
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