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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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity
import Mathlib.Analysis.SpecialFunctions.Exp
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionEntropyAlgebra
open MatrixMultiplication.Foundation
def binaryEntropy (a : ℝ) : ℝ := entropyTerm a + entropyTerm (1 - a)
def conditionalMixEntropy (a : ℝ) (m : ℕ) (Hc Ho : ℝ) : ℝ :=
((m : ℝ) * (binaryEntropy a + a * Hc + (1 - a) * Ho) -
binaryEntropy (a ^ m) - a ^ m * ((m : ℝ) * Hc)) / (1 - a ^ m)
theorem conditionalMixEntropy_eq_split (a : ℝ) (m : ℕ) (Hc Ho : ℝ)
(hden : 1 - a ^ m ≠ 0) :
conditionalMixEntropy a m Hc Ho =
((m : ℝ) * (binaryEntropy a + a * Hc + (1 - a) * Ho) -
entropyTerm (a ^ m) - a ^ m * ((m : ℝ) * Hc)) / (1 - a ^ m) +
Real.log (1 - a ^ m) := by
dsimp [conditionalMixEntropy, binaryEntropy, entropyTerm]
field_simp
ring
private theorem conditionalMixEntropy_cancel
(a C O L hc ho : ℝ) (m : ℕ)
(hden : 1 - a ^ m ≠ 0)
(hcLog : Real.log a = C - L)
(hoLog : Real.log (1 - a) = O - L) :
conditionalMixEntropy a m (C + hc) (O + ho) =
m * L + Real.log (1 - a ^ m) +
m * ((a - a ^ m) / (1 - a ^ m) * hc +
(1 - a) / (1 - a ^ m) * ho) := by
dsimp [conditionalMixEntropy, binaryEntropy, entropyTerm]
rw [Real.log_pow, hcLog, hoLog]
field_simp
ring
private theorem ratio_range (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
0 < x / (x + y) ∧ x / (x + y) < 1 := by
have hs : 0 < x + y := add_pos hx hy
exact ⟨div_pos hx hs, (div_lt_one hs).mpr (by linarith)⟩
private theorem ratio_complement (x y : ℝ) (h : x + y ≠ 0) :
1 - x / (x + y) = y / (x + y) := by
field_simp
exact add_sub_cancel_left x y
theorem binaryEntropy_cardinal_mixture (np nq : ℝ) (hnp : 0 < np) (hnq : 0 < nq) :
let a := np / (np + nq)
binaryEntropy a + a * Real.log np + (1 - a) * Real.log nq =
Real.log (np + nq) := by
let a := np / (np + nq)
have hs : 0 < np + nq := add_pos hnp hnq
have hcLog : Real.log a = Real.log np - Real.log (np + nq) :=
Real.log_div hnp.ne' hs.ne'
have hoLog : Real.log (1 - a) = Real.log nq - Real.log (np + nq) := by
rw [show 1 - a = nq / (np + nq) from ratio_complement np nq hs.ne',
Real.log_div hnq.ne' hs.ne']
change binaryEntropy a + a * Real.log np + (1 - a) * Real.log nq = _
dsimp [binaryEntropy, entropyTerm]
rw [hcLog, hoLog]
ring
theorem conditionalMixEntropy_incident
(np nq hc ho : ℝ) (m : ℕ)
(hnp : 0 < np) (hnq : 0 < nq) (hm : 0 < m) :
let a := np / (np + nq)
conditionalMixEntropy a m (Real.log np + hc) (Real.log nq + ho) -
Real.log ((np + nq) ^ m - np ^ m) =
m * ((a - a ^ m) / (1 - a ^ m) * hc +
(1 - a) / (1 - a ^ m) * ho) := by
let a := np / (np + nq)
change conditionalMixEntropy a m (Real.log np + hc) (Real.log nq + ho) -
Real.log ((np + nq) ^ m - np ^ m) = _
have hs : 0 < np + nq := add_pos hnp hnq
have ha : 0 < a ∧ a < 1 := ratio_range np nq hnp hnq
have hden : 0 < 1 - a ^ m :=
sub_pos.mpr (pow_lt_one₀ ha.1.le ha.2 (Nat.ne_of_gt hm))
have hcLog : Real.log a = Real.log np - Real.log (np + nq) :=
Real.log_div hnp.ne' hs.ne'
have hoLog : Real.log (1 - a) = Real.log nq - Real.log (np + nq) := by
rw [show 1 - a = nq / (np + nq) from ratio_complement np nq hs.ne',
Real.log_div hnq.ne' hs.ne']
have hfactor : (np + nq) ^ m - np ^ m =
(np + nq) ^ m * (1 - a ^ m) := by
dsimp [a]
rw [div_pow]
field_simp
have hlog : Real.log ((np + nq) ^ m - np ^ m) =
m * Real.log (np + nq) + Real.log (1 - a ^ m) := by
rw [hfactor, Real.log_mul (pow_ne_zero _ hs.ne') hden.ne', Real.log_pow]
rw [conditionalMixEntropy_cancel a (Real.log np) (Real.log nq)
(Real.log (np + nq)) hc ho m hden.ne' hcLog hoLog, hlog]
ring
theorem conditionalMixEntropy_gibbs (q hc ho : ℝ) (m : ℕ) (hm : 0 < m) :
conditionalMixEntropy (Real.exp hc / (Real.exp hc + Real.exp ho)) m
(Real.log q + hc) (Real.log q + ho) - m * Real.log q =
Real.log ((Real.exp hc + Real.exp ho) ^ m - Real.exp ((m : ℝ) * hc)) := by
let a := Real.exp hc / (Real.exp hc + Real.exp ho)
have ha : 0 < a ∧ a < 1 := ratio_range _ _ (Real.exp_pos hc) (Real.exp_pos ho)
have hden : 1 - a ^ m ≠ 0 :=
(sub_pos.mpr (pow_lt_one₀ ha.1.le ha.2 (Nat.ne_of_gt hm))).ne'
have hmean : m * ((a - a ^ m) / (1 - a ^ m) * Real.log q +
(1 - a) / (1 - a ^ m) * Real.log q) = m * Real.log q := by
field_simp
ring
have h := conditionalMixEntropy_incident (Real.exp hc) (Real.exp ho)
(Real.log q) (Real.log q) m (Real.exp_pos hc) (Real.exp_pos ho) hm
change conditionalMixEntropy a m (Real.log (Real.exp hc) + Real.log q)
(Real.log (Real.exp ho) + Real.log q) -
Real.log ((Real.exp hc + Real.exp ho) ^ m - Real.exp hc ^ m) =
m * ((a - a ^ m) / (1 - a ^ m) * Real.log q +
(1 - a) / (1 - a ^ m) * Real.log q) at h
rw [Real.log_exp, Real.log_exp, add_comm hc (Real.log q),
add_comm ho (Real.log q), hmean, ← Real.exp_nat_mul hc m] at h
change conditionalMixEntropy a m (Real.log q + hc) (Real.log q + ho) -
m * Real.log q = _
linarith
end MatrixMultiplication.CompletionEntropyAlgebra
end
end OAI
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