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cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Populations
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldTerminalPopulations
open AllFieldHistory
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K : ℕ} {H : Type*} [Fintype H]
{A : H → Type*} [∀ h, Fintype (A h)]
def law (ν : ∀ h, A h → ℚ) (x : Sigma A) : ℚ := ν x.1 x.2
def denominator (ν : ∀ h, A h → ℚ) : ℕ :=
AllFieldPopulationCounts.denominator (law ν)
def dilation (ν : ∀ h, A h → ℚ) (m : ℕ) : ℕ := m * denominator ν
def count (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (m : ℕ) (h : H) (a : A h) : ℕ :=
population allocation m (history h) *
AllFieldPopulationCounts.baseCount (law ν) ⟨h, a⟩
theorem denominator_pos (ν : ∀ h, A h → ℚ) : 0 < denominator ν :=
AllFieldPopulationCounts.denominator_pos (law ν)
theorem dilation_pos (ν : ∀ h, A h → ℚ) {m : ℕ} (hm : 0 < m) :
0 < dilation ν m := Nat.mul_pos hm (denominator_pos ν)
theorem baseCount_cast (ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(h : H) (a : A h) :
(AllFieldPopulationCounts.baseCount (law ν) ⟨h, a⟩ : ℚ) =
(denominator ν : ℚ) * ν h a :=
AllFieldPopulationCounts.baseCount_cast (law ν) (fun x => hν x.1 x.2) ⟨h, a⟩
theorem sum_baseCount (ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) (h : H) :
(∑ a, AllFieldPopulationCounts.baseCount (law ν) ⟨h, a⟩) = denominator ν := by
apply Nat.cast_injective (R := ℚ)
simp only [Nat.cast_sum, baseCount_cast ν hν, ← Finset.mul_sum, htotal, mul_one]
theorem population_dilation (allocation : Allocation) (ν : ∀ h, A h → ℚ)
(m : ℕ) (h : History K) :
population allocation (dilation ν m) h =
population allocation m h * denominator ν := by
simp only [population, AllFieldPopulationCounts.count, dilation]
ac_rfl
theorem populationLength_dilation (allocation : Allocation) (ν : ∀ h, A h → ℚ)
(m : ℕ) :
populationLength (K := K) allocation (dilation ν m) =
populationLength (K := K) allocation m * denominator ν := by
simp only [populationLength, AllFieldPopulationCounts.blockLength, dilation]
ac_rfl
theorem sum_count (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) (m : ℕ) (h : H) :
(∑ a, count allocation history ν m h a) =
population allocation (dilation ν m) (history h) := by
simp only [count, ← Finset.mul_sum, sum_baseCount ν hν htotal,
population_dilation]
theorem count_cast (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(m : ℕ) (h : H) (a : A h) :
(count allocation history ν m h a : ℚ) =
(populationLength (K := K) allocation (dilation ν m) : ℚ) *
amount allocation (history h) * ν h a := by
rw [count, Nat.cast_mul, population_cast, baseCount_cast ν hν,
populationLength_dilation, Nat.cast_mul]
ring
theorem count_cast_real (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(m : ℕ) (h : H) (a : A h) :
(count allocation history ν m h a : ℝ) =
(populationLength (K := K) allocation (dilation ν m) : ℝ) *
(amount allocation (history h) : ℝ) * (ν h a : ℝ) := by
exact_mod_cast count_cast allocation history ν hν m h a
theorem count_dilate (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (m n : ℕ) (h : H) (a : A h) :
count allocation history ν (m * n) h a =
m * count allocation history ν n h a := by
simp only [count, population, AllFieldPopulationCounts.count]
ac_rfl
theorem count_mul_one (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (m : ℕ) (h : H) (a : A h) :
count allocation history ν m h a = m * count allocation history ν 1 h a := by
simpa only [Nat.mul_one] using count_dilate allocation history ν m 1 h a
theorem source_population_pos (allocation : Allocation) (ν : ∀ h, A h → ℚ)
{m : ℕ} (hm : 0 < m) (h : History K) (hh : 0 < amount allocation h) :
0 < population allocation (dilation ν m) h :=
(AllFieldPopulationCounts.count_pos_iff (amount allocation)
(amount_nonneg allocation) (dilation_pos ν hm) h).2 hh
theorem sum_count_pos (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) {m : ℕ} (hm : 0 < m)
(h : H) (hh : 0 < amount allocation (history h)) :
0 < ∑ a, count allocation history ν m h a := by
rw [sum_count allocation history ν hν htotal]
exact source_population_pos allocation ν hm (history h) hh
theorem count_positive_law_positive (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(m : ℕ) (h : H) (a : A h) (ha : 0 < count allocation history ν m h a) :
0 < ν h a := by
have hb : 0 < AllFieldPopulationCounts.baseCount (law ν) ⟨h, a⟩ :=
Nat.pos_of_mul_pos_left ha
have hbq : (0 : ℚ) < AllFieldPopulationCounts.baseCount (law ν) ⟨h, a⟩ := by
exact_mod_cast hb
rw [baseCount_cast ν hν] at hbq
have hD : (0 : ℚ) < denominator ν := by exact_mod_cast denominator_pos ν
exact (mul_pos_iff_of_pos_left hD).mp hbq
theorem count_pos_iff (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
{m : ℕ} (hm : 0 < m) (h : H) (hh : 0 < amount allocation (history h))
(a : A h) : 0 < count allocation history ν m h a ↔ 0 < ν h a := by
constructor
· exact count_positive_law_positive allocation history ν hν m h a
· intro ha
apply Nat.mul_pos
· exact (AllFieldPopulationCounts.count_pos_iff (amount allocation)
(amount_nonneg allocation) hm (history h)).2 hh
· have hb := (AllFieldPopulationCounts.count_pos_iff (law ν)
(fun x => hν x.1 x.2) (m := 1) (by decide) ⟨h, a⟩).2 ha
simpa only [AllFieldPopulationCounts.count, one_mul] using hb
theorem count_eq_zero_of_law_eq_zero (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(m : ℕ) (h : H) (a : A h) (ha : ν h a = 0) :
count allocation history ν m h a = 0 := by
apply Nat.cast_injective (R := ℚ)
simp only [count_cast allocation history ν hν, ha, mul_zero, Nat.cast_zero]
theorem count_ratio (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) {m : ℕ} (hm : 0 < m)
(h : H) (hh : 0 < amount allocation (history h)) (a : A h) :
(count allocation history ν m h a : ℚ) /
(∑ b, count allocation history ν m h b : ℕ) = ν h a := by
rw [sum_count allocation history ν hν htotal, count_cast allocation history ν hν,
population_cast]
have hp : (populationLength (K := K) allocation (dilation ν m) : ℚ) *
amount allocation (history h) ≠ 0 := by
rw [← population_cast]
exact_mod_cast (Nat.ne_of_gt (source_population_pos allocation ν hm (history h) hh))
exact mul_div_cancel_left₀ _ hp
theorem count_ratio_real (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) {m : ℕ} (hm : 0 < m)
(h : H) (hh : 0 < amount allocation (history h)) (a : A h) :
(count allocation history ν m h a : ℝ) /
(∑ b, count allocation history ν m h b : ℕ) = (ν h a : ℝ) := by
have hq := congrArg (fun q : ℚ => (q : ℝ))
(count_ratio allocation history ν hν htotal hm h hh a)
simpa only [Rat.cast_div, Rat.cast_natCast] using hq
theorem normalizedCountLaw_mass (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) {m : ℕ} (hm : 0 < m)
(h : H) (hh : 0 < amount allocation (history h)) (a : A h) :
(ConditionalLabels.normalizedCountLaw (count allocation history ν m h)
(sum_count_pos allocation history ν hν htotal hm h hh)).mass a = (ν h a : ℝ) :=
count_ratio_real allocation history ν hν htotal hm h hh a
theorem normalizedCountLaw_mass_eq (allocation : Allocation) (history : H → History K)
(ν : ∀ h, A h → ℚ) (hν : ∀ h a, 0 ≤ ν h a)
(htotal : ∀ h, ∑ a, ν h a = 1) {m : ℕ} (hm : 0 < m)
(h : H) (hh : 0 < amount allocation (history h)) :
(ConditionalLabels.normalizedCountLaw (count allocation history ν m h)
(sum_count_pos allocation history ν hν htotal hm h hh)).mass =
fun a => (ν h a : ℝ) :=
funext (normalizedCountLaw_mass allocation history ν hν htotal hm h hh)
end MatrixMultiplication.AllFieldTerminalPopulations
end
end OAI
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