openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /MarginalSupport.lean
| 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 | |