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import OAI.LinearAlgebra.MatrixMultiplication.Duality.EntropyHelpers
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualInformation
open MatrixMultiplication.Foundation DualEntropyHelpers
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
universe u v w
variable {A : Type u} [Fintype A] {X : Type w} [Fintype X]
def ConditionalIndependent {R L : Type*} [Fintype R] [Fintype L]
(p : FiniteLaw A) (prior : A → R) (label : A → L) (x : A → X) : Prop :=
∀ r, 0 < (p.map prior).mass r → ∀ l xx,
((p.conditional prior r).map (fun a => (label a, x a))).mass (l, xx) =
((p.conditional prior r).map label).mass l *
((p.conditional prior r).map x).mass xx
def ConditionalMassFactorization {R L : Type*} [Fintype R] [Fintype L]
(p : FiniteLaw A) (prior : A → R) (label : A → L) (x : A → X) : Prop :=
∀ r l xx,
(p.map (fun a => (prior a, (label a, x a)))).mass (r, (l, xx)) *
(p.map prior).mass r =
(p.map (fun a => (prior a, label a))).mass (r, l) *
(p.map (fun a => (prior a, x a))).mass (r, xx)
theorem conditionalIndependent_of_mass_factorization {R L : Type*}
[Fintype R] [Fintype L] (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X)
(factor : ConditionalMassFactorization p prior label x) :
ConditionalIndependent p prior label x := by
intro r hr l xx
have h := factor r l xx
rw [← p.map_mass_mul_conditional_map prior (fun a => (label a, x a)) r (l, xx),
← p.map_mass_mul_conditional_map prior label r l,
← p.map_mass_mul_conditional_map prior x r xx] at h
apply mul_left_cancel₀ (mul_ne_zero hr.ne' hr.ne')
calc
((p.map prior).mass r * (p.map prior).mass r) *
((p.conditional prior r).map (fun a => (label a, x a))).mass (l, xx) =
((p.map prior).mass r *
((p.conditional prior r).map (fun a => (label a, x a))).mass (l, xx)) *
(p.map prior).mass r := by ring
_ = ((p.map prior).mass r * ((p.conditional prior r).map label).mass l) *
((p.map prior).mass r * ((p.conditional prior r).map x).mass xx) := h
_ = ((p.map prior).mass r * (p.map prior).mass r) *
(((p.conditional prior r).map label).mass l *
((p.conditional prior r).map x).mass xx) := by ring
theorem mass_factorization_of_conditionalIndependent {R L : Type*}
[Fintype R] [Fintype L] (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X)
(independent : ConditionalIndependent p prior label x) :
ConditionalMassFactorization p prior label x := by
intro r l xx
rw [← p.map_mass_mul_conditional_map prior (fun a => (label a, x a)) r (l, xx),
← p.map_mass_mul_conditional_map prior label r l,
← p.map_mass_mul_conditional_map prior x r xx]
by_cases hr : (p.map prior).mass r = 0
· simp only [hr, zero_mul, mul_zero]
· have hpos : 0 < (p.map prior).mass r :=
lt_of_le_of_ne ((p.map prior).nonneg r) (Ne.symm hr)
rw [independent r hpos l xx]
ring
theorem conditionalIndependent_iff_mass_factorization {R L : Type*}
[Fintype R] [Fintype L] (p : FiniteLaw A)
(prior : A → R) (label : A → L) (x : A → X) :
ConditionalIndependent p prior label x ↔ ConditionalMassFactorization p prior label x :=
⟨mass_factorization_of_conditionalIndependent p prior label x,
conditionalIndependent_of_mass_factorization p prior label x⟩
theorem joint_append_readable_entropy {R L : Type*} [Fintype R] [Fintype L]
(p : FiniteLaw A) (prior : A → R) (label : A → L) (x : A → X)
(read : R × X → L) (hread : ∀ a, read (prior a, x a) = label a) :
finiteEntropy (p.map (fun a => ((prior a, label a), x a))).mass =
finiteEntropy (p.map (fun a => (prior a, x a))).mass := by
let lift : R × X → (R × L) × X := fun rx => ((rx.1, read rx), rx.2)
have hinj : Function.Injective lift := by
intro a b h
apply Prod.ext
· exact congrArg (fun t : (R × L) × X => t.1.1) h
· exact congrArg (fun t : (R × L) × X => t.2) h
have hcomp : (fun a => lift (prior a, x a)) =
(fun a => ((prior a, label a), x a)) := by
funext a
simp only [lift, hread]
have hm := map_comp_mass p (fun a => (prior a, x a)) lift
rw [hcomp] at hm
calc
finiteEntropy (p.map (fun a => ((prior a, label a), x a))).mass =
finiteEntropy (((p.map (fun a => (prior a, x a))).map lift).mass) := by
rw [hm]
_ = _ := (p.map (fun a => (prior a, x a))).map_entropy_of_injective lift hinj
variable {Label : ℕ → Type v} [∀ n, Fintype (Label n)]
def remainingEntropy (p : FiniteLaw A) (labels : ∀ n, A → Label n)
(x : A → X) (n : ℕ) : ℝ :=
finiteEntropy (p.map (fun a => (labelRecordOf labels n a, x a))).mass -
finiteEntropy (p.map (labelRecordOf labels n)).mass
@[simp] theorem remainingEntropy_zero (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (x : A → X) :
remainingEntropy p labels x 0 = finiteEntropy (p.map x).mass := by
have hm := map_comp_mass p x (fun xx : X => ((PUnit.unit : PUnit.{v + 1}), xx))
have he : finiteEntropy (p.map (fun a => (labelRecordOf labels 0 a, x a))).mass =
finiteEntropy (p.map x).mass := by
change finiteEntropy (p.map (fun a => ((PUnit.unit : PUnit.{v + 1}), x a))).mass =
finiteEntropy (p.map x).mass
rw [← hm]
exact (p.map x).map_entropy_of_injective _ (fun _ _ h => congrArg Prod.snd h)
have hz : finiteEntropy (p.map (labelRecordOf labels 0)).mass = 0 := by
change finiteEntropy (p.map (fun _ : A => (PUnit.unit : PUnit.{v + 1}))).mass = 0
simp [finiteEntropy, FiniteLaw.map_mass, p.total, entropyTerm]
rw [remainingEntropy, he, hz, sub_zero]
theorem remainingEntropy_complete (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (x : A → X) (depth : ℕ)
(complete : Function.Injective (labelRecordOf labels depth)) :
remainingEntropy p labels x depth = 0 := by
have hi : Function.Injective (fun a => (labelRecordOf labels depth a, x a)) :=
fun _ _ h => complete (congrArg Prod.fst h)
rw [remainingEntropy, p.map_entropy_of_injective _ hi,
p.map_entropy_of_injective _ complete, sub_self]
theorem prefix_entropy_succ (p : FiniteLaw A) (labels : ∀ n, A → Label n)
(depth n : ℕ) (hn : n < depth) :
finiteEntropy (p.map (labelRecordOf labels (n + 1))).mass =
finiteEntropy (p.map (labelRecordOf labels n)).mass +
(FiniteLabelHierarchy.ofLawLabels p labels depth).refinementEntropy n := by
have h := (FiniteLabelHierarchy.ofLawLabels p labels depth).prefixLaw_entropy_succ
n (Nat.succ_le_of_lt hn)
rw [FiniteLabelHierarchy.ofLawLabels_prefixLaw_mass,
FiniteLabelHierarchy.ofLawLabels_prefixLaw_mass] at h
exact h
theorem remainingEntropy_succ_readable (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (x : A → X) (depth n : ℕ) (hn : n < depth)
(read : LabelRecord Label n × X → Label n)
(hread : ∀ a, read (labelRecordOf labels n a, x a) = labels n a) :
remainingEntropy p labels x n - remainingEntropy p labels x (n + 1) =
(FiniteLabelHierarchy.ofLawLabels p labels depth).refinementEntropy n := by
have hj := joint_append_readable_entropy p (labelRecordOf labels n) (labels n)
x read hread
have hp := prefix_entropy_succ p labels depth n hn
change finiteEntropy (p.map (fun a => (labelRecordOf labels (n + 1) a, x a))).mass =
finiteEntropy (p.map (fun a => (labelRecordOf labels n a, x a))).mass at hj
unfold remainingEntropy
rw [hj, hp]
ring
theorem remainingEntropy_succ_independent (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (x : A → X) (n : ℕ)
(independent : ConditionalIndependent p (labelRecordOf labels n) (labels n) x) :
remainingEntropy p labels x (n + 1) = remainingEntropy p labels x n := by
exact conditional_independence_entropy p (labelRecordOf labels n) (labels n) x independent
theorem incident_rate_eq_coordinate_entropy (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (depth : ℕ)
(reader : Fin depth → ReaderPair) (x : A → X)
(complete : Function.Injective (labelRecordOf labels depth))
(readable : ∀ n : Fin depth, reader n ≠ .yz →
∃ read : LabelRecord Label n.val × X → Label n.val,
∀ a, read (labelRecordOf labels n.val a, x a) = labels n.val a)
(independent : ∀ n : Fin depth, reader n = .yz →
ConditionalIndependent p (labelRecordOf labels n.val) (labels n.val) x) :
ConditionalLabels.lawPairingRate p labels depth reader .xy +
ConditionalLabels.lawPairingRate p labels depth reader .xz =
finiteEntropy (p.map x).mass := by
let E := remainingEntropy p labels x
have hterm (n : Fin depth) :
(if reader n = .xy then
(FiniteLabelHierarchy.ofLawLabels p labels depth).refinementEntropy n.val else 0) +
(if reader n = .xz then
(FiniteLabelHierarchy.ofLawLabels p labels depth).refinementEntropy n.val else 0) =
E n.val - E (n.val + 1) := by
by_cases hi : reader n = .yz
· have he := remainingEntropy_succ_independent p labels x n.val (independent n hi)
simp only [hi, reduceCtorEq, ite_false]
change 0 + 0 = remainingEntropy p labels x n.val -
remainingEntropy p labels x (n.val + 1)
rw [he]
ring
· obtain ⟨read, hread⟩ := readable n hi
have he := remainingEntropy_succ_readable p labels x depth n.val n.isLt read hread
change _ = remainingEntropy p labels x n.val -
remainingEntropy p labels x (n.val + 1)
rw [he]
cases hr : reader n <;> simp_all
have htel : ∀ k, (∑ n ∈ Finset.range k, (E n - E (n + 1))) = E 0 - E k := by
intro k
induction k with
| zero => simp
| succ k ih => rw [Finset.sum_range_succ, ih]; ring
calc
ConditionalLabels.lawPairingRate p labels depth reader .xy +
ConditionalLabels.lawPairingRate p labels depth reader .xz =
∑ n : Fin depth, (E n.val - E (n.val + 1)) := by
simp only [ConditionalLabels.lawPairingRate, ConditionalLabels.pairingRate,
← Finset.sum_add_distrib]
exact Finset.sum_congr rfl (fun n _ => hterm n)
_ = E 0 - E depth := by
calc
(∑ n : Fin depth, (E n.val - E (n.val + 1))) =
∑ n ∈ Finset.range depth, (E n - E (n + 1)) :=
Fin.sum_univ_eq_sum_range (fun n => E n - E (n + 1)) depth
_ = E 0 - E depth := htel depth
_ = finiteEntropy (p.map x).mass := by
rw [show E 0 = finiteEntropy (p.map x).mass from remainingEntropy_zero p labels x,
show E depth = 0 from remainingEntropy_complete p labels x depth complete, sub_zero]
theorem nonincident_rate_eq_entropy_sub_coordinate (p : FiniteLaw A)
(labels : ∀ n, A → Label n) (depth : ℕ)
(reader : Fin depth → ReaderPair) (x : A → X)
(complete : Function.Injective (labelRecordOf labels depth))
(readable : ∀ n : Fin depth, reader n ≠ .yz →
∃ read : LabelRecord Label n.val × X → Label n.val,
∀ a, read (labelRecordOf labels n.val a, x a) = labels n.val a)
(independent : ∀ n : Fin depth, reader n = .yz →
ConditionalIndependent p (labelRecordOf labels n.val) (labels n.val) x) :
ConditionalLabels.lawPairingRate p labels depth reader .yz =
finiteEntropy p.mass - finiteEntropy (p.map x).mass := by
have hall := ConditionalLabels.lawPairingRate_sum p labels depth reader complete
have hx := incident_rate_eq_coordinate_entropy p labels depth reader x
complete readable independent
linarith
end MatrixMultiplication.DualInformation
end
end OAI
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