| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Extraction.JointMaskedSelection |
| import OAI.LinearAlgebra.MatrixFields.Extraction.JointPairClassWindows |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| 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 |
|
|
| 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 |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
