File size: 2,634 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AmbientDegree
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointMarginalSupport
open MatrixMultiplication.Foundation JointPopulation JointAmbientDegree
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ) (sum : H → ℕ)
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_coordinate_witness (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (s : Fin 3) (i : Positions counts h) :
∃ j : Positions counts h,
shapeSide s ((canonicalTarget counts h).val j) = shapeSide s (w i) := by
have hp : 0 < wordPopulation (shapeSide s ∘ w) (shapeSide s (w i)) := by
unfold wordPopulation
exact Fintype.card_pos_iff.mpr ⟨⟨i, rfl⟩⟩
rw [hw.2 s (shapeSide s (w i))] at hp
obtain ⟨⟨j, hj⟩⟩ := Fintype.card_pos_iff.mp hp
exact ⟨j, hj⟩
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_coordinate_support (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (allowed : Fin 3 → Fin 17 → Prop)
(hallowed : ∀ u, 0 < counts h u → ∀ s, allowed s (shapeSide s u)) :
∀ i s, allowed s (shapeSide s (w i)) := by
intro i s
obtain ⟨j, hj⟩ := localAllowed_coordinate_witness counts sum h w hw s i
rw [← hj]
exact hallowed ((canonicalTarget counts h).val j)
(symbol_count_pos counts (canonicalTarget counts) h j) s
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_parent_bounds (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (parent : Fin 3 → ℕ)
(hparent : ∀ u, 0 < counts h u → ∀ s, (shapeSide s u).val ≤ parent s) :
∀ i s, (shapeSide s (w i)).val ≤ parent s :=
localAllowed_coordinate_support counts sum h w hw
(fun s a => a.val ≤ parent s) hparent
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_positive_reference (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (parent : Fin 3 → ℕ) (p : Shape → ℝ)
(hparent : ∀ u, 0 < counts h u → ∀ s, (shapeSide s u).val ≤ parent s)
(hp : ∀ u : Shape, u.1.val + u.2.1.val + u.2.2.val = sum h →
(∀ s, (shapeSide s u).val ≤ parent s) → 0 < p u) :
∀ i, 0 < p (w i) := by
intro i
exact hp (w i) (hw.1 i) (localAllowed_parent_bounds counts sum h w hw parent hparent i)
end MatrixMultiplication.JointMarginalSupport
end
end OAI
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