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f0f05de | 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 | import Mathlib.Data.Fintype.Pi
import Mathlib.Probability.Distributions.Uniform
namespace OAI
namespace Balanced.InjectionExposure
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
noncomputable def completions (K : Finset I) (base : I ↪ J) : Finset (I ↪ J) :=
Finset.univ.filter (fun e => ∀ j ∈ K, e j = base j)
def remaining (K : Finset I) (base : I ↪ J) : Finset J :=
Finset.univ \ K.image base
@[simp] theorem mem_completions (K : Finset I) (base e : I ↪ J) :
e ∈ completions K base ↔ ∀ j ∈ K, e j = base j := by
simp [completions]
omit [Fintype I] [DecidableEq I] in
@[simp] theorem mem_remaining (K : Finset I) (base : I ↪ J) (a : J) :
a ∈ remaining K base ↔ a ∉ K.image base := by
simp [remaining]
theorem completions_nonempty (K : Finset I) (base : I ↪ J) :
(completions K base).Nonempty := by
exact ⟨base, by simp⟩
theorem value_mem_remaining (K : Finset I) (base e : I ↪ J)
(he : e ∈ completions K base) (i : I) (hi : i ∉ K) :
e i ∈ remaining K base := by
rw [mem_remaining]
intro h
obtain ⟨j, hj, hji⟩ := Finset.mem_image.mp h
have hsame : e j = e i := (mem_completions K base e).mp he j hj |>.trans hji
exact hi (e.injective hsame ▸ hj)
theorem base_mem_remaining (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) : base i ∈ remaining K base :=
value_mem_remaining K base base (by simp) i hi
omit [Fintype I] [DecidableEq I] in
theorem card_remaining (K : Finset I) (base : I ↪ J) :
(remaining K base).card = Fintype.card J - K.card := by
rw [remaining, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ,
Finset.card_image_of_injective _ base.injective]
theorem swap_mem_completions (K : Finset I) (base e : I ↪ J)
(he : e ∈ completions K base) (a b : J)
(ha : a ∈ remaining K base) (hb : b ∈ remaining K base) :
e.trans (Equiv.swap a b).toEmbedding ∈ completions K base := by
rw [mem_completions] at he ⊢
intro j hj
change Equiv.swap a b (e j) = base j
rw [he j hj]
apply Equiv.swap_apply_of_ne_of_ne
· intro h
exact (mem_remaining K base a).mp ha (Finset.mem_image.mpr ⟨j, hj, h⟩)
· intro h
exact (mem_remaining K base b).mp hb (Finset.mem_image.mpr ⟨j, hj, h⟩)
theorem fibre_card_eq (K : Finset I) (base : I ↪ J) (i : I)
(a b : J) (ha : a ∈ remaining K base) (hb : b ∈ remaining K base) :
((completions K base).filter (fun e => e i = a)).card =
((completions K base).filter (fun e => e i = b)).card := by
let swapOutput : (I ↪ J) → (I ↪ J) :=
fun e => e.trans (Equiv.swap a b).toEmbedding
have hinv (e : I ↪ J) : swapOutput (swapOutput e) = e := by
ext j
exact Equiv.swap_apply_self a b (e j)
have hmap (e : I ↪ J)
(he : e ∈ (completions K base).filter (fun e => e i = a)) :
swapOutput e ∈ (completions K base).filter (fun e => e i = b) := by
obtain ⟨he, hei⟩ := Finset.mem_filter.mp he
refine Finset.mem_filter.mpr ⟨swap_mem_completions K base e he a b ha hb, ?_⟩
change Equiv.swap a b (e i) = b
rw [hei, Equiv.swap_apply_left]
have hmap' (e : I ↪ J)
(he : e ∈ (completions K base).filter (fun e => e i = b)) :
swapOutput e ∈ (completions K base).filter (fun e => e i = a) := by
obtain ⟨he, hei⟩ := Finset.mem_filter.mp he
refine Finset.mem_filter.mpr ⟨swap_mem_completions K base e he a b ha hb, ?_⟩
change Equiv.swap a b (e i) = a
rw [hei, Equiv.swap_apply_right]
exact Finset.card_bij' (fun e _ => swapOutput e) (fun e _ => swapOutput e)
hmap hmap' (fun e _ => hinv e) (fun e _ => hinv e)
theorem card_completions_eq (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (a : J) (ha : a ∈ remaining K base) :
(completions K base).card = (remaining K base).card *
((completions K base).filter (fun e => e i = a)).card := by
calc
(completions K base).card =
∑ b ∈ remaining K base,
((completions K base).filter (fun e => e i = b)).card :=
Finset.card_eq_sum_card_fiberwise (fun e he =>
value_mem_remaining K base e he i hi)
_ = ∑ _b ∈ remaining K base,
((completions K base).filter (fun e => e i = a)).card := by
apply Finset.sum_congr rfl
intro b hb
exact fibre_card_eq K base i b a hb ha
_ = _ := by simp
theorem next_value_probability (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (a : J) (ha : a ∈ remaining K base) :
((PMF.uniformOfFinset (completions K base) (completions_nonempty K base)).toOuterMeasure
{e | e i = a}).toReal = 1 / ((remaining K base).card : ℝ) := by
classical
have hC : 0 < ((completions K base).card : ℝ) := by
exact_mod_cast Finset.card_pos.mpr (completions_nonempty K base)
have hR : 0 < ((remaining K base).card : ℝ) := by
exact_mod_cast Finset.card_pos.mpr (show (remaining K base).Nonempty from ⟨a, ha⟩)
have hfactor : ((completions K base).card : ℝ) =
((remaining K base).card : ℝ) *
(((completions K base).filter (fun e => e i = a)).card : ℝ) := by
exact_mod_cast card_completions_eq K base i hi a ha
rw [PMF.toOuterMeasure_uniformOfFinset_apply]
simp only [ENNReal.toReal_div, ENNReal.toReal_natCast, Set.mem_ofPred_eq]
apply (div_eq_div_iff (ne_of_gt hC) (ne_of_gt hR)).mpr
nlinarith
theorem next_value_probability_card (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (a : J) (ha : a ∈ remaining K base) :
((PMF.uniformOfFinset (completions K base) (completions_nonempty K base)).toOuterMeasure
{e | e i = a}).toReal = 1 / ((Fintype.card J - K.card : ℕ) : ℝ) := by
rw [next_value_probability K base i hi a ha, card_remaining]
theorem event_card_eq (K : Finset I) (base : I ↪ J) (i : I) (hi : i ∉ K)
(P : J → Prop) [DecidablePred P] :
((completions K base).filter (fun e => P (e i))).card =
((remaining K base).filter P).card *
((completions K base).filter (fun e => e i = base i)).card := by
calc
((completions K base).filter (fun e => P (e i))).card =
∑ b ∈ (remaining K base).filter P,
(((completions K base).filter (fun e => P (e i))).filter
(fun e => e i = b)).card := by
apply Finset.card_eq_sum_card_fiberwise
intro e he
obtain ⟨he, hPe⟩ := Finset.mem_filter.mp he
exact Finset.mem_filter.mpr ⟨value_mem_remaining K base e he i hi, hPe⟩
_ = ∑ _b ∈ (remaining K base).filter P,
((completions K base).filter (fun e => e i = base i)).card := by
apply Finset.sum_congr rfl
intro b hb
obtain ⟨hb, hPb⟩ := Finset.mem_filter.mp hb
have hfibre : ((completions K base).filter (fun e => P (e i))).filter
(fun e => e i = b) = (completions K base).filter (fun e => e i = b) := by
ext e
simp only [Finset.mem_filter]
constructor
· exact fun h => ⟨h.1.1, h.2⟩
· intro h
exact ⟨⟨h.1, h.2 ▸ hPb⟩, h.2⟩
rw [hfibre]
exact fibre_card_eq K base i b (base i) hb (base_mem_remaining K base i hi)
_ = _ := by simp
theorem next_event_probability (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (P : J → Prop) [DecidablePred P] :
((PMF.uniformOfFinset (completions K base) (completions_nonempty K base)).toOuterMeasure
{e | P (e i)}).toReal =
(((remaining K base).filter P).card : ℝ) / ((remaining K base).card : ℝ) := by
classical
have hC : 0 < ((completions K base).card : ℝ) := by
exact_mod_cast Finset.card_pos.mpr (completions_nonempty K base)
have hR : 0 < ((remaining K base).card : ℝ) := by
exact_mod_cast Finset.card_pos.mpr
(show (remaining K base).Nonempty from ⟨base i, base_mem_remaining K base i hi⟩)
have hfactor : ((completions K base).card : ℝ) =
((remaining K base).card : ℝ) *
(((completions K base).filter (fun e => e i = base i)).card : ℝ) := by
exact_mod_cast card_completions_eq K base i hi (base i) (base_mem_remaining K base i hi)
have hevent : (((completions K base).filter (fun e => P (e i))).card : ℝ) =
(((remaining K base).filter P).card : ℝ) *
(((completions K base).filter (fun e => e i = base i)).card : ℝ) := by
exact_mod_cast event_card_eq K base i hi P
rw [PMF.toOuterMeasure_uniformOfFinset_apply]
simp only [ENNReal.toReal_div, ENNReal.toReal_natCast, Set.mem_ofPred_eq]
apply (div_eq_div_iff (ne_of_gt hC) (ne_of_gt hR)).mpr
rw [hfactor, hevent]
ring
theorem next_event_probability_le (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (P : J → Prop) [DecidablePred P]
(hP : ((remaining K base).filter P).card ≤ 1) :
((PMF.uniformOfFinset (completions K base) (completions_nonempty K base)).toOuterMeasure
{e | P (e i)}).toReal ≤ 1 / ((remaining K base).card : ℝ) := by
rw [next_event_probability K base i hi P]
exact div_le_div_of_nonneg_right (by exact_mod_cast hP) (Nat.cast_nonneg _)
theorem next_event_probability_le_budget (K : Finset I) (base : I ↪ J)
(i : I) (hi : i ∉ K) (P : J → Prop) [DecidablePred P]
(hP : ((remaining K base).filter P).card ≤ 1)
(N : ℕ) (hK : K.card ≤ N) (hN : N < Fintype.card J) :
((PMF.uniformOfFinset (completions K base) (completions_nonempty K base)).toOuterMeasure
{e | P (e i)}).toReal ≤ 1 / ((Fintype.card J - N : ℕ) : ℝ) := by
apply (next_event_probability_le K base i hi P hP).trans
rw [card_remaining]
apply one_div_le_one_div_of_le
· exact_mod_cast Nat.sub_pos_of_lt hN
· exact_mod_cast Nat.sub_le_sub_left hK (Fintype.card J)
end Balanced.InjectionExposure
end OAI
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