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import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AssignmentLoss
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.ExtractionRates
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointMaskedSelection
open JointCoarseHashing JointExtractionRates
attribute [local instance] Classical.propDecidable
variable {P E O V : Type*}
structure OrbitData (P E O V : Type*) where
ambient : Finset (Triple P)
support : P → ℕ
coarse : E → Triple P
support_ambient : ∀ f ∈ ambient, HasSupportSum support f
target_mem : ∀ e, coarse e ∈ ambient
orbits : E → Finset O
full : E → O → Finset V
passing : E → O → Finset V
full_pos : ∀ e, ∀ o ∈ orbits e, 0 < (full e o).card
passing_subset : ∀ e, ∀ o ∈ orbits e, passing e o ⊆ full e o
competitors : E → O → V → Finset (Triple P)
competitor_mem : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o,
∀ f ∈ competitors e o v, f ∈ ambient
competitor_ne : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o,
∀ f ∈ competitors e o v, f ≠ coarse e
competitor_shares : ∀ e, ∀ o ∈ orbits e, ∀ v ∈ passing e o,
∀ f ∈ competitors e o v, SharesSide (coarse e) f
namespace OrbitData
def eligibilityCompetitors (d : OrbitData P E O V) (e : E) : Finset (Triple P) :=
d.ambient.filter (fun f => f ≠ d.coarse e ∧ f.1 = (d.coarse e).1)
@[simp] theorem mem_eligibilityCompetitors (d : OrbitData P E O V) (e : E) (f : Triple P) :
f ∈ d.eligibilityCompetitors e ↔
f ∈ d.ambient ∧ f ≠ d.coarse e ∧ f.1 = (d.coarse e).1 := by
simp only [eligibilityCompetitors, Finset.mem_filter]
section Finite
variable [Fintype P]
def variableBad (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(e : E) (o : O) (v : V) (s : Sample P k) : Prop :=
∃ f ∈ d.competitors e o v, Survives k U f s
def eligibilityGood (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(e : E) (s : Sample P k) : Prop :=
¬ ∃ f ∈ d.eligibilityCompetitors e, Survives k U f s
def Failure (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(N : ℝ) (e : E) (s : Sample P k) : Prop :=
JointLossCounts.targetBad (d.orbits e) (d.full e) (d.passing e)
(d.variableBad k U e) (d.eligibilityGood k U e) N s
def Good (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(N : ℝ) (e : E) (s : Sample P k) : Prop :=
Survives k U (d.coarse e) s ∧ ¬ d.Failure k U N e s
def goodEvent (d : OrbitData P E O V) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(N : ℝ) (e : E) : Finset (Sample P k) :=
(ownEvent k U (d.coarse e)).filter (fun s => ¬ d.Failure k U N e s)
@[simp] theorem mem_goodEvent (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k) :
s ∈ d.goodEvent k U N e ↔ d.Good k U N e s := by
simp only [goodEvent, ownEvent, Finset.mem_filter, Finset.mem_univ, true_and, Good]
theorem variableBad_card_le (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (e : E) (o : O) (ho : o ∈ d.orbits e)
(v : V) (hv : v ∈ d.passing e o) (D : ℕ)
(hdegree : (d.competitors e o v).card ≤ D) :
(((ownEvent k U (d.coarse e)).filter (d.variableBad k U e o v)).card : ℝ) ≤
((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) * ((ownEvent k U (d.coarse e)).card : ℝ) := by
have h := competitor_union_card_le k U d.support (d.coarse e)
(d.support_ambient _ (d.target_mem e)) (d.competitors e o v)
(fun f hf => d.support_ambient f (d.competitor_mem e o ho v hv f hf))
(d.competitor_ne e o ho v hv) (d.competitor_shares e o ho v hv)
have hdegreeR : ((d.competitors e o v).card : ℝ) ≤ (D : ℝ) := by
exact_mod_cast hdegree
have hfilter : (ownEvent k U (d.coarse e)).filter (d.variableBad k U e o v) =
(ownEvent k U (d.coarse e)).filter
(fun s => ∃ f ∈ d.competitors e o v, Survives k U f s) := by
ext s
simp only [Finset.mem_filter, variableBad]
rw [hfilter]
exact h.trans (mul_le_mul_of_nonneg_right
(div_le_div_of_nonneg_right hdegreeR (Nat.cast_nonneg _))
(Nat.cast_nonneg _))
theorem eligibilityFailure_card_le (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (e : E) :
(((ownEvent k U (d.coarse e)).filter (fun s => ¬ d.eligibilityGood k U e s)).card : ℝ) ≤
(((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ)) *
((ownEvent k U (d.coarse e)).card : ℝ) := by
have h := competitor_union_card_le k U d.support (d.coarse e)
(d.support_ambient _ (d.target_mem e)) (d.eligibilityCompetitors e)
(fun f hf => d.support_ambient f ((d.mem_eligibilityCompetitors e f).1 hf).1)
(fun f hf => ((d.mem_eligibilityCompetitors e f).1 hf).2.1)
(fun f hf => Or.inl ((d.mem_eligibilityCompetitors e f).1 hf).2.2)
simpa only [eligibilityGood, not_not] using h
theorem failure_card_le (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (hN : 0 < N) (e : E) (D : ℕ)
(hdegree : ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o, (d.competitors e o v).card ≤ D) :
(((ownEvent k U (d.coarse e)).filter (d.Failure k U N e)).card : ℝ) ≤
(((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) +
((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ))) *
((ownEvent k U (d.coarse e)).card : ℝ) := by
exact JointLossCounts.targetBad_card_le (ownEvent k U (d.coarse e))
(d.orbits e) (d.full e) (d.passing e) (d.variableBad k U e)
(d.eligibilityGood k U e) ((D : ℝ) / ((37 ^ k : ℕ) : ℝ))
(((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ)) N
(div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) hN (d.full_pos e)
(d.passing_subset e)
(fun o ho v hv => d.variableBad_card_le k U e o ho v hv D (hdegree o ho v hv))
(d.eligibilityFailure_card_le k U e)
theorem ownEvent_card_real (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (e : E) :
((ownEvent k U (d.coarse e)).card : ℝ) =
((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2) * (Fintype.card (Sample P k) : ℝ) := by
have hsample : (Fintype.card (Sample P k) : ℝ) ≠ 0 := by
exact_mod_cast Fintype.card_ne_zero
have hf : ((ownEvent k U (d.coarse e)).card : ℝ) / (Fintype.card (Sample P k) : ℝ) =
(U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 := by
change
((Finset.univ.filter (fun s =>
JointHashing.Survives (U : Set (ZMod (37 ^ k)))
(castWord k (d.coarse e).1) (castWord k (d.coarse e).2.1) s)).card : ℝ) /
(Fintype.card (JointHashing.Sample P (ZMod (37 ^ k))) : ℝ) = _
simpa only [Fintype.card_subtype, ZMod.card] using
JointHashing.survival_fraction U (castWord k (d.coarse e).1) (castWord k (d.coarse e).2.1)
exact (div_eq_iff hsample).mp hf
theorem goodEvent_card_lower (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (hN : 0 < N) (e : E) (D : ℕ)
(hdegree : ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o, (d.competitors e o v).card ≤ D)
(hsmall : ((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) +
((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) ≤ 1 / 2) :
((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) *
(Fintype.card (Sample P k) : ℝ) ≤ (d.goodEvent k U N e).card := by
have hb : (((ownEvent k U (d.coarse e)).filter (d.Failure k U N e)).card : ℝ) ≤
(1 / 2) * ((ownEvent k U (d.coarse e)).card : ℝ) :=
(d.failure_card_le k U N hN e D hdegree).trans
(mul_le_mul_of_nonneg_right hsmall (Nat.cast_nonneg _))
have hg := JointLossCounts.good_card_lower (ownEvent k U (d.coarse e))
(d.Failure k U N e) (1 / 2) hb
rw [d.ownEvent_card_real] at hg
change _ ≤ ((d.goodEvent k U N e).card : ℝ) at hg
convert hg using 1
ring
theorem goodIncidence_eq (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) :
goodIncidence (fun s e => d.Good k U N e s) e = (d.goodEvent k U N e).card := by
unfold goodIncidence
congr 1
ext s
simp only [Finset.mem_filter, Finset.mem_univ, true_and, mem_goodEvent]
theorem good_eligibility (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (e : E) (s : Sample P k)
(hg : d.Good k U N e s) : d.eligibilityGood k U e s := by
by_contra h
exact hg.2 (Or.inl h)
theorem good_no_shared_x (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (N : ℝ) (e f : E) (s : Sample P k)
(he : d.Good k U N e s) (hf : d.Good k U N f s)
(hne : d.coarse f ≠ d.coarse e) : (d.coarse f).1 ≠ (d.coarse e).1 := by
intro hx
exact d.good_eligibility k U N e s he
⟨d.coarse f, (d.mem_eligibilityCompetitors e _).2 ⟨d.target_mem f, hne, hx⟩, hf.1⟩
theorem good_missingFraction_le (d : OrbitData P E O V) (k : ℕ)
(U : Finset (ZMod (37 ^ k))) (C₀ N : ℝ) (e : E) (s : Sample P k)
(hg : d.Good k U N e s)
(hdet : ∀ o ∈ d.orbits e,
JointLossCounts.deterministicLossFraction (d.full e o) (d.passing e o) ≤ C₀ / N)
(o : O) (ho : o ∈ d.orbits e) :
JointLossCounts.missingFraction (d.full e o) (d.passing e o)
(d.variableBad k U e o) s ≤ (C₀ + 1) / N :=
JointLossCounts.good_missingFraction_le (d.orbits e) (d.full e) (d.passing e)
(d.variableBad k U e) (d.eligibilityGood k U e) C₀ N s hg.2
(d.passing_subset e) hdet o ho
theorem exists_selection [Fintype E] (d : OrbitData P E O V)
(hinj : Function.Injective d.coarse) (k : ℕ) (U : Finset (ZMod (37 ^ k)))
(C₀ N : ℝ) (hN : 0 < N) (D K : ℕ)
(hdegree : ∀ e, ∀ o ∈ d.orbits e, ∀ v ∈ d.passing e o,
(d.competitors e o v).card ≤ D)
(hsmall : ∀ e, ((d.eligibilityCompetitors e).card : ℝ) / ((37 ^ k : ℕ) : ℝ) +
((d.orbits e).card : ℝ) * N * ((D : ℝ) / ((37 ^ k : ℕ) : ℝ)) ≤ 1 / 2)
(hdet : ∀ e, ∀ o ∈ d.orbits e,
JointLossCounts.deterministicLossFraction (d.full e o) (d.passing e o) ≤ C₀ / N)
(hK : (K : ℝ) ≤ (Fintype.card E : ℝ) *
((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2)) :
∃ (s : Sample P k) (G : Finset E), G.card = K ∧
(∀ e ∈ G, d.Good k U N e s) ∧
(∀ e ∈ G, ∀ f ∈ G, e ≠ f → (d.coarse e).1 ≠ (d.coarse f).1) ∧
∀ e ∈ G, ∀ o ∈ d.orbits e,
JointLossCounts.missingFraction (d.full e o) (d.passing e o)
(d.variableBad k U e o) s ≤ (C₀ + 1) / N := by
classical
have hi (e : E) :
((U.card : ℝ) / ((37 ^ k : ℕ) : ℝ) ^ 2 / 2) *
(Fintype.card (Sample P k) : ℝ) ≤
(goodIncidence (fun s e => d.Good k U N e s) e : ℝ) := by
rw [d.goodIncidence_eq]
exact d.goodEvent_card_lower k U N hN e D (hdegree e) (hsmall e)
obtain ⟨s, hs⟩ := exists_goodCount_ge (fun s e => d.Good k U N e s) _ hi
have hk : K ≤ goodCount (fun s e => d.Good k U N e s) s := by exact_mod_cast hK.trans hs
obtain ⟨G, hcard, hgood⟩ := exists_good_subfamily (fun s e => d.Good k U N e s) s K hk
refine ⟨s, G, hcard, hgood, ?_, ?_⟩
· intro e he f hf hne
exact d.good_no_shared_x k U N f e s (hgood f hf) (hgood e he)
(fun h => hne (hinj h))
· intro e he o ho
exact d.good_missingFraction_le k U C₀ N e s (hgood e he) (hdet e) o ho
end Finite
end OrbitData
end MatrixMultiplication.JointMaskedSelection
end
end OAI
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