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f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CoarseHashing
import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexTypeCounting
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointPopulation
open MatrixMultiplication.Foundation
open scoped BigOperators
attribute [local instance] Classical.propDecidable
structure HistoryKey where
lot : ℕ
initialShape : Fin 3 → ℕ
physicalPlacement : Equiv.Perm (Fin 3)
parentSplits : List ((Fin 3 → ℕ) × Bool)
subdivisions : List ℕ
abbrev Shape := Fin 17 × Fin 17 × Fin 17
def shapeSide (s : Fin 3) (u : Shape) : Fin 17 :=
if s = 0 then u.1 else if s = 1 then u.2.1 else u.2.2
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ)
abbrev Positions (h : H) := Fin (∑ u, counts h u)
abbrev Target := ∀ h, ExactWords (counts h)
def canonicalTarget : Target counts :=
fun h => Classical.choice (exactWords_nonempty (counts h))
theorem target_card : Fintype.card (Target counts) =
∏ h, Nat.multinomial Finset.univ (counts h) := by
simp only [Target, Fintype.card_pi, exactWords_card]
omit [Fintype H] [DecidableEq H] in
theorem target_nonempty : Nonempty (Target counts) := ⟨canonicalTarget counts⟩
abbrev Position := Σ h, Positions counts h
def triple (e : Target counts) : JointCoarseHashing.Triple (Position counts) :=
(fun p => (e p.1).val p.2 |>.1,
fun p => (e p.1).val p.2 |>.2.1,
fun p => (e p.1).val p.2 |>.2.2)
omit [Fintype H] [DecidableEq H] in
theorem triple_injective : Function.Injective (triple counts) := by
intro e f h
funext c
apply Subtype.ext
funext i
apply Prod.ext
· exact congrArg (fun t : JointCoarseHashing.Triple (Position counts) => t.1 ⟨c, i⟩) h
· apply Prod.ext
· exact congrArg (fun t : JointCoarseHashing.Triple (Position counts) => t.2.1 ⟨c, i⟩) h
· exact congrArg (fun t : JointCoarseHashing.Triple (Position counts) => t.2.2 ⟨c, i⟩) h
def targetSet : Finset (JointCoarseHashing.Triple (Position counts)) :=
Finset.univ.image (triple counts)
theorem targetSet_card : (targetSet counts).card =
∏ h, Nat.multinomial Finset.univ (counts h) := by
rw [targetSet, Finset.card_image_of_injective _ (triple_injective counts)]
simpa using target_card counts
abbrev Class (e : Target counts) (h : H) (u : Shape) :=
{i : Positions counts h // (e h).val i = u}
omit [Fintype H] [DecidableEq H] in
theorem class_card (e : Target counts) (h : H) (u : Shape) :
Fintype.card (Class counts e h u) = counts h u := (e h).property u
omit [Fintype H] [DecidableEq H] in
theorem symbol_count_pos (e : Target counts) (h : H) (i : Positions counts h) :
0 < counts h ((e h).val i) := by
rw [← class_card counts e h ((e h).val i)]
exact Fintype.card_pos_iff.mpr ⟨⟨i, rfl⟩⟩
omit [Fintype H] [DecidableEq H] in
theorem triple_support (sum : H → ℕ)
(hsupport : ∀ h u, 0 < counts h u →
u.1.val + u.2.1.val + u.2.2.val = sum h) (e : Target counts) :
JointCoarseHashing.HasSupportSum (fun p : Position counts => sum p.1) (triple counts e) := by
intro p
exact hsupport p.1 ((e p.1).val p.2) (symbol_count_pos counts e p.1 p.2)
omit [Fintype H] [DecidableEq H] in
theorem exists_position_permutation (e f : Target counts) :
∃ p : ∀ h, Equiv.Perm (Positions counts h), ∀ h, (e h).val ∘ p h = (f h).val := by
have hp : ∀ h, ∃ p : Equiv.Perm (Positions counts h), (e h).val ∘ p = (f h).val := by
intro h
apply (samePopulation_iff_permutation _ _).mp
intro u
rw [(e h).property, (f h).property]
choose p hp using hp
exact ⟨p, hp⟩
def sideWord (s : Fin 3) (t : JointCoarseHashing.Triple (Position counts))
(h : H) : Positions counts h → Fin 17 :=
fun i => shapeSide s (t.1 ⟨h, i⟩, t.2.1 ⟨h, i⟩, t.2.2 ⟨h, i⟩)
omit [Fintype H] [DecidableEq H] in
theorem sideWord_triple (s : Fin 3) (e : Target counts) (h : H) :
sideWord counts s (triple counts e) h = shapeSide s ∘ (e h).val := rfl
omit [Fintype H] [DecidableEq H] in
theorem marginal_population_eq (e f : Target counts) (h : H) (s : Fin 3) (a : Fin 17) :
wordPopulation (sideWord counts s (triple counts e) h) a =
wordPopulation (sideWord counts s (triple counts f) h) a := by
obtain ⟨p, hp⟩ := exists_position_permutation counts e f
rw [sideWord_triple, sideWord_triple, ← hp h]
exact (wordPopulation_reindex (shapeSide s ∘ (e h).val) (p h) a).symm
def ambientSet (sum : H → ℕ) : Finset (JointCoarseHashing.Triple (Position counts)) :=
Finset.univ.filter fun t =>
JointCoarseHashing.HasSupportSum (fun p : Position counts => sum p.1) t ∧
∀ h s a, wordPopulation (sideWord counts s t h) a =
wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a
theorem targetSet_subset_ambient (sum : H → ℕ)
(hsupport : ∀ h u, 0 < counts h u →
u.1.val + u.2.1.val + u.2.2.val = sum h) :
targetSet counts ⊆ ambientSet counts sum := by
intro t ht
obtain ⟨e, _, rfl⟩ := Finset.mem_image.mp ht
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ _, triple_support counts sum hsupport e, ?_⟩
intro h s a
exact marginal_population_eq counts e (canonicalTarget counts) h s a
end MatrixMultiplication.JointPopulation
end
end OAI
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