openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CanonicalMixed.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalCW | |
| import Mathlib.Tactic.FinCases | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointCanonicalMixed | |
| open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW | |
| open CWStrands HistorySymmetry InheritedMasks | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {H F : Type*} [Fintype H] [DecidableEq H] [CommRing F] | |
| (counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ) | |
| (parentShape : H → Fin 3 → ℕ) (sharing : H → Bool) | |
| (hleft : ∀ h, leftLength h ≤ 8) | |
| def inputTensor (h : H) : Tensor F | |
| (Left leftLength h × Right rightLength h) | |
| (Left leftLength h × Right rightLength h) | |
| (Left leftLength h × Right rightLength h) := | |
| if sharing h then parentTensor leftLength rightLength parentShape h | |
| else Tensor.product (strand (Fin (leftLength h))) (strand (Fin (rightLength h))) | |
| def childRight (c : ClassKey (H := H)) : | |
| Tensor F (Right rightLength c.1) (Right rightLength c.1) (Right rightLength c.1) := | |
| if sharing c.1 then rightTensor rightLength parentShape c | |
| else strand (Fin (rightLength c.1)) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem block_eq_children (h : H) (u : Shape) | |
| (hu : sharing h = true → ∀ s, shapeNat u s ≤ parentShape h s) : | |
| block (Left leftLength) (Right rightLength) | |
| (inputTensor leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) (h, u) = | |
| Tensor.product (leftTensor (F := F) leftLength (h, u)) | |
| (childRight rightLength parentShape sharing (h, u)) := by | |
| by_cases hs : sharing h = true | |
| · funext x y z | |
| have hh := congrFun (congrFun (congrFun | |
| (block_eq_product (F := F) leftLength rightLength parentShape hleft h u (hu hs)) x) y) z | |
| simpa only [block, inputTensor, childRight, hs, ↓reduceIte] using hh | |
| · funext x y z | |
| simp [block, inputTensor, childRight, hs, leftTensor, shapeTensor, | |
| coarse, CWCoarseIndices.coarseIndex, Fin.ext_iff, shapeNat, shapeSide, | |
| Tensor.product, ite_mul] | |
| omit [DecidableEq H] in | |
| theorem canonicalBase_eq_children | |
| (hsupport : ∀ h u, 0 < counts h u → sharing h = true → | |
| ∀ s, shapeNat u s ≤ parentShape h s) : | |
| canonicalBase counts (Left leftLength) (Right rightLength) | |
| (inputTensor leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) = | |
| pairClassProduct (leftTensor (F := F) leftLength) | |
| (childRight rightLength parentShape sharing) := by | |
| funext x y z | |
| unfold canonicalBase | |
| have hf : (∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c, | |
| block (Left leftLength) (Right rightLength) | |
| (inputTensor leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) c | |
| (x.left c j, x.right c j) (y.left c j, y.right c j) (z.left c j, z.right c j)) = | |
| ∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c, | |
| (leftTensor (F := F) leftLength c (x.left c j) (y.left c j) (z.left c j) * | |
| childRight rightLength parentShape sharing c | |
| (x.right c j) (y.right c j) (z.right c j)) := by | |
| apply Finset.prod_congr rfl | |
| intro c _ | |
| apply Finset.prod_congr rfl | |
| intro j _ | |
| have hc : 0 < counts c.1 c.2 := lt_of_le_of_lt (Nat.zero_le j.val) j.isLt | |
| rw [block_eq_children leftLength rightLength parentShape sharing hleft c.1 c.2 | |
| (hsupport c.1 c.2 hc)] | |
| rfl | |
| rw [hf] | |
| simp only [pairClassProduct, classProduct, Finset.prod_mul_distrib] | |
| def rawSource : Tensor F (RawStrands counts leftLength rightLength) | |
| (RawStrands counts leftLength rightLength) (RawStrands counts leftLength rightLength) := | |
| classProduct (fun h => if sharing h then | |
| shapeTensor (Fin (leftLength h) ⊕ Fin (rightLength h)) (parentShape h) | |
| else strand (Fin (leftLength h) ⊕ Fin (rightLength h))) | |
| theorem inputTensor_restrict : | |
| Tensor.restrict (joinMatrix (F := F) counts leftLength rightLength) | |
| (joinMatrix counts leftLength rightLength) (joinMatrix counts leftLength rightLength) | |
| (rawSource counts leftLength rightLength parentShape sharing) = | |
| sourceTensor counts (Left leftLength) (Right rightLength) | |
| (inputTensor leftLength rightLength parentShape sharing) := by | |
| classical | |
| funext x y z | |
| simp only [Tensor.restrict, joinMatrix] | |
| simp only [ite_mul, mul_ite, zero_mul, mul_zero, one_mul, Finset.sum_ite_eq', | |
| Finset.mem_univ, ↓reduceIte] | |
| unfold rawSource sourceTensor classProduct inputTensor | |
| apply Finset.prod_congr rfl | |
| intro h _ | |
| apply Finset.prod_congr rfl | |
| intro i _ | |
| by_cases hs : sharing h = true | |
| · simp [hs, joinWords, parentTensor] | |
| · simp [hs, joinWords, strand_sum_elim] | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem coarse_support (h : H) | |
| (x y z : Left leftLength h × Right rightLength h) | |
| (hT : inputTensor (F := F) leftLength rightLength parentShape sharing h x y z ≠ 0) : | |
| (coarse leftLength rightLength hleft 0 h x).val + | |
| (coarse leftLength rightLength hleft 1 h y).val + | |
| (coarse leftLength rightLength hleft 2 h z).val = 2 * leftLength h := by | |
| by_cases hs : sharing h = true | |
| · have hp : parentTensor (F := F) leftLength rightLength parentShape h x y z ≠ 0 := by | |
| simpa [inputTensor, hs] using hT | |
| have hh := CWCoarseIndices.split_coarse_support | |
| (F := F) (by simpa using hleft h) (parentShape h) x y z hp | |
| simpa only [coarse, Fintype.card_fin] using hh.1 | |
| · have ht : strand (F := F) (Fin (leftLength h)) x.1 y.1 z.1 ≠ 0 := by | |
| intro hz | |
| apply hT | |
| simp [inputTensor, hs, Tensor.product, hz] | |
| simpa only [coarse, Fintype.card_fin] using | |
| CWCoarseIndices.strand_coarse_support (F := F) (by simpa using hleft h) x.1 y.1 z.1 ht | |
| theorem typeWindow_zero_one {P A : Type*} [Fintype P] [DecidableEq P] | |
| [Fintype A] [DecidableEq A] (w : P → A) : typeWindow (fun _ => 0) 1 w := by | |
| intro a | |
| dsimp [empiricalLaw] | |
| have h0 : (0 : ℝ) ≤ (wordPopulation w a : ℝ) / Fintype.card P := by positivity | |
| rw [sub_zero, abs_of_nonneg h0] | |
| by_cases hp : Fintype.card P = 0 | |
| · simp [hp] | |
| · apply (div_le_one (by exact_mod_cast Nat.pos_of_ne_zero hp)).2 | |
| exact_mod_cast (Fintype.card_subtype_le (fun i : P => w i = a)) | |
| abbrev MixedWords := RawPairs counts (Left leftLength) (Right rightLength) | |
| def coarseWord (w : MixedWords counts leftLength rightLength) : Position counts → Fin 17 := | |
| JointCanonicalization.coarseWord counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 0 w | |
| def sideMask (s : Fin 3) (w : MixedWords counts leftLength rightLength) : Prop := | |
| ∀ h a, wordPopulation (fun i => coarseWord counts leftLength rightLength hleft w ⟨h, i⟩) a = | |
| wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a | |
| def maskedSource : Tensor F (MixedWords counts leftLength rightLength) | |
| (MixedWords counts leftLength rightLength) (MixedWords counts leftLength rightLength) := | |
| ExactRecovery.delete | |
| (sourceTensor counts (Left leftLength) (Right rightLength) | |
| (inputTensor leftLength rightLength parentShape sharing)) | |
| (sideMask counts leftLength rightLength hleft 0) | |
| (sideMask counts leftLength rightLength hleft 1) | |
| (sideMask counts leftLength rightLength hleft 2) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem sideMask_of_target (s : Fin 3) (e : Target counts) | |
| (w : MixedWords counts leftLength rightLength) | |
| (he : coarseWord counts leftLength rightLength hleft w = | |
| fun i => sideWord counts s (triple counts e) i.1 i.2) : | |
| sideMask counts leftLength rightLength hleft s w := by | |
| intro h a | |
| rw [he] | |
| exact marginal_population_eq counts e (canonicalTarget counts) h s a | |
| theorem maskedSource_support (x y z : MixedWords counts leftLength rightLength) | |
| (hT : maskedSource (F := F) counts leftLength rightLength parentShape sharing hleft x y z ≠ 0) : | |
| (coarseWord counts leftLength rightLength hleft x, | |
| coarseWord counts leftLength rightLength hleft y, | |
| coarseWord counts leftLength rightLength hleft z) ∈ | |
| ambientSet counts (fun h => 2 * leftLength h) := by | |
| have hm : sideMask counts leftLength rightLength hleft 0 x ∧ | |
| sideMask counts leftLength rightLength hleft 1 y ∧ | |
| sideMask counts leftLength rightLength hleft 2 z := by | |
| by_contra hn | |
| simp [maskedSource, ExactRecovery.delete, hn] at hT | |
| have hr : sourceTensor counts (Left leftLength) (Right rightLength) | |
| (inputTensor (F := F) leftLength rightLength parentShape sharing) x y z ≠ 0 := by | |
| simpa [maskedSource, ExactRecovery.delete, hm] using hT | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_univ _, ?_, ?_⟩ | |
| · intro i | |
| have hi : inputTensor (F := F) leftLength rightLength parentShape sharing i.1 | |
| (x i.1 i.2) (y i.1 i.2) (z i.1 i.2) ≠ 0 := by | |
| intro hz | |
| apply hr | |
| unfold sourceTensor classProduct | |
| apply Finset.prod_eq_zero (Finset.mem_univ i.1) | |
| exact Finset.prod_eq_zero (Finset.mem_univ i.2) hz | |
| exact coarse_support leftLength rightLength parentShape sharing hleft i.1 | |
| (x i.1 i.2) (y i.1 i.2) (z i.1 i.2) hi | |
| · intro h s a | |
| fin_cases s | |
| · refine Eq.trans ?_ (hm.1 h a) | |
| apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a) | |
| funext i | |
| simp [sideWord, shapeSide] | |
| · refine Eq.trans ?_ (hm.2.1 h a) | |
| apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a) | |
| funext i | |
| simp [sideWord, shapeSide] | |
| · refine Eq.trans ?_ (hm.2.2 h a) | |
| apply congrArg (fun v : Positions counts h → Fin 17 => wordPopulation v a) | |
| funext i | |
| simp [sideWord, shapeSide] | |
| omit [DecidableEq H] in | |
| theorem coarseIdeal_support_masks (e : Target counts) | |
| (x y z : MixedWords counts leftLength rightLength) | |
| (hT : coarseIdeal counts (Left leftLength) (Right rightLength) | |
| (inputTensor (F := F) leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) e x y z ≠ 0) : | |
| sideMask counts leftLength rightLength hleft 0 x ∧ | |
| sideMask counts leftLength rightLength hleft 1 y ∧ | |
| sideMask counts leftLength rightLength hleft 2 z := by | |
| have hc : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 0 e x ∧ | |
| coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 1 e y ∧ | |
| coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 2 e z := by | |
| by_contra hn | |
| simp [coarseIdeal, ExactRecovery.delete, hn] at hT | |
| constructor | |
| · apply sideMask_of_target counts leftLength rightLength hleft 0 e x | |
| funext i | |
| exact hc.1 i.1 i.2 | |
| constructor | |
| · apply sideMask_of_target counts leftLength rightLength hleft 1 e y | |
| funext i | |
| exact hc.2.1 i.1 i.2 | |
| · apply sideMask_of_target counts leftLength rightLength hleft 2 e z | |
| funext i | |
| exact hc.2.2 i.1 i.2 | |
| omit [DecidableEq H] in | |
| theorem ideal_support_masks (e : Target counts) | |
| (px py pz : MixedWords counts leftLength rightLength → Prop) | |
| (x y z : MixedWords counts leftLength rightLength) | |
| (hT : ExactRecovery.delete | |
| (coarseIdeal counts (Left leftLength) (Right rightLength) | |
| (inputTensor (F := F) leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) e) px py pz x y z ≠ 0) : | |
| sideMask counts leftLength rightLength hleft 0 x ∧ | |
| sideMask counts leftLength rightLength hleft 1 y ∧ | |
| sideMask counts leftLength rightLength hleft 2 z := by | |
| apply coarseIdeal_support_masks (F := F) counts leftLength rightLength parentShape sharing hleft e x y z | |
| intro hz | |
| simp [ExactRecovery.delete, hz] at hT | |
| omit [DecidableEq H] in | |
| theorem delete_ideal_eq (e : Target counts) | |
| (px py pz : MixedWords counts leftLength rightLength → Prop) : | |
| let Q := ExactRecovery.delete | |
| (coarseIdeal counts (Left leftLength) (Right rightLength) | |
| (inputTensor (F := F) leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) e) px py pz | |
| ExactRecovery.delete Q (sideMask counts leftLength rightLength hleft 0) | |
| (sideMask counts leftLength rightLength hleft 1) | |
| (sideMask counts leftLength rightLength hleft 2) = Q := by | |
| dsimp only | |
| funext x y z | |
| by_cases h : ExactRecovery.delete | |
| (coarseIdeal counts (Left leftLength) (Right rightLength) | |
| (inputTensor (F := F) leftLength rightLength parentShape sharing) | |
| (coarse leftLength rightLength hleft) e) px py pz x y z = 0 | |
| · rw [ExactRecovery.delete, h] | |
| simp | |
| · have hm := ideal_support_masks (F := F) counts leftLength rightLength parentShape sharing hleft | |
| e px py pz x y z h | |
| rw [ExactRecovery.delete, ite_eq_left hm] | |
| end MatrixMultiplication.JointCanonicalMixed | |
| end | |
| end OAI | |