openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /PopulationMixedCompatibility.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationCompatibility | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationMixedNonzero | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWStrandCompatibility | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointPopulationMixedCompatibility | |
| open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW | |
| open JointPopulationCompatibility JointPopulationMixedNonzero CWStrands InheritedMasks | |
| open CWStrandCompatibility | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| def sharingDesignated {H : Type*} (sharing : H → Bool) | |
| (d : JointPopulationCompatibility.ClassKey (H := H) → Shape → Prop) | |
| (c : JointPopulationCompatibility.ClassKey (H := H)) (u : Shape) : Prop := | |
| sharing c.1 = true ∧ d c u | |
| variable {H F : Type*} [Fintype H] [DecidableEq H] [Field F] | |
| (counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ) | |
| (parentShape : H → Fin 3 → ℕ) (sharing : H → Bool) (hleft : ∀ h, leftLength h ≤ 8) | |
| (hsupport : ∀ h u, 0 < counts h u → sharing h = true → ∀ s, shapeNat u s ≤ parentShape h s) | |
| (e : Target counts) | |
| (x y z : RawPairs counts (Left leftLength) (Right rightLength)) | |
| (hsource : sourceTensor counts (Left leftLength) (Right rightLength) | |
| (JointCanonicalMixed.inputTensor (F := F) leftLength rightLength parentShape sharing) x y z ≠ 0) | |
| (hx : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 0 e x) | |
| (hy : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 1 e y) | |
| (hz : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 2 e z) | |
| variable (SL SR : H → Type*) | |
| [∀ h, Fintype (SL h)] [∀ h, DecidableEq (SL h)] | |
| [∀ h, Fintype (SR h)] [∀ h, DecidableEq (SR h)] | |
| (leftStatistic : ∀ h, Left leftLength h → SL h) | |
| (rightStatistic : ∀ h, Right rightLength h → SR h) | |
| (κLeft : ∀ h, SL h ≃ SL h) (κRight : ∀ h, SR h ≃ SR h) | |
| (hstatLeft : ∀ h w, leftStatistic h (CWCompatibilityTransfer.wordComplement w) = | |
| κLeft h (leftStatistic h w)) | |
| (hstatRight : ∀ h w, rightStatistic h (CWCompatibilityTransfer.wordComplement w) = | |
| κRight h (rightStatistic h w)) | |
| (xLeftLaw yLeftLaw : ∀ c : JointCanonicalization.ClassKey (H := H), SL c.1 → ℝ) | |
| (xRightLaw yRightLaw : ∀ c : JointCanonicalization.ClassKey (H := H), SR c.1 → ℝ) | |
| (χ : ℝ) | |
| include hsource hx hy hz hsupport hstatLeft hstatRight | |
| omit [DecidableEq H] in | |
| theorem source_implies_compatibleY | |
| (leftLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SL c.1 → ℝ) | |
| (rightLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SR c.1 → ℝ) | |
| (hlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedYLeft (h, shapeSide 1 u) u → | |
| leftLaw (h, shapeSide 1 u) u = xLeftLaw (h, u) ∘ (κLeft h).symm) | |
| (hrlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedYRight parentShape (h, shapeSide 1 u) u → | |
| rightLaw (h, shapeSide 1 u) u = xRightLaw (h, u) ∘ (κRight h).symm) | |
| (huseX : ∀ h u, 0 < counts h u → sharing h = true → | |
| typeWindow (xLeftLaw (h, u)) χ | |
| (fun i : Class counts e h u => leftStatistic h (x h i.val).1) ∧ | |
| typeWindow (xRightLaw (h, u)) χ | |
| (fun i : Class counts e h u => rightStatistic h (x h i.val).2)) : | |
| Compatible counts (Left leftLength) (Right rightLength) | |
| (fun c => SL c.1) (fun c => SR c.1) | |
| (fun c => leftStatistic c.1) (fun c => rightStatistic c.1) | |
| (sharingDesignated sharing designatedYLeft) | |
| (sharingDesignated sharing (designatedYRight parentShape)) leftLaw rightLaw χ 1 e y := by | |
| apply compatible_of_designated_windows | |
| · intro h u hn hd | |
| rw [hlaw h u hn hd.1 hd.2] | |
| exact zeroZ_transfer (fun _ : Class counts e h u => shapeNat u) | |
| (fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1) | |
| (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft | |
| hsupport e x y z hsource hx hy hz h u hn hd.1).1 | |
| (leftStatistic h) (κLeft h) (hstatLeft h) (xLeftLaw (h, u)) χ | |
| (fun _ => hd.2) (huseX h u hn hd.1).1 | |
| · intro h u hn hd | |
| rw [hrlaw h u hn hd.1 hd.2] | |
| exact zeroZ_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u) | |
| (fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2) | |
| (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft | |
| hsupport e x y z hsource hx hy hz h u hn hd.1).2 | |
| (rightStatistic h) (κRight h) (hstatRight h) (xRightLaw (h, u)) χ | |
| (fun _ => hd.2) (huseX h u hn hd.1).2 | |
| omit [DecidableEq H] in | |
| theorem source_implies_compatibleZ | |
| (leftLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SL c.1 → ℝ) | |
| (rightLaw : ∀ c : JointPopulationCompatibility.ClassKey (H := H), Shape → SR c.1 → ℝ) | |
| (hlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedZLeft (h, shapeSide 2 u) u → | |
| leftLaw (h, shapeSide 2 u) u = | |
| (if shapeNat u 1 = 0 then xLeftLaw (h, u) else yLeftLaw (h, u)) ∘ | |
| (κLeft h).symm) | |
| (hrlaw : ∀ h u, 0 < counts h u → sharing h = true → designatedZRight parentShape (h, shapeSide 2 u) u → | |
| rightLaw (h, shapeSide 2 u) u = | |
| (if (parentShape h - shapeNat u) 1 = 0 then xRightLaw (h, u) | |
| else yRightLaw (h, u)) ∘ (κRight h).symm) | |
| (huseX : ∀ h u, 0 < counts h u → sharing h = true → | |
| typeWindow (xLeftLaw (h, u)) χ | |
| (fun i : Class counts e h u => leftStatistic h (x h i.val).1) ∧ | |
| typeWindow (xRightLaw (h, u)) χ | |
| (fun i : Class counts e h u => rightStatistic h (x h i.val).2)) | |
| (huseY : ∀ h u, 0 < counts h u → sharing h = true → | |
| typeWindow (yLeftLaw (h, u)) χ | |
| (fun i : Class counts e h u => leftStatistic h (y h i.val).1) ∧ | |
| typeWindow (yRightLaw (h, u)) χ | |
| (fun i : Class counts e h u => rightStatistic h (y h i.val).2)) : | |
| Compatible counts (Left leftLength) (Right rightLength) | |
| (fun c => SL c.1) (fun c => SR c.1) | |
| (fun c => leftStatistic c.1) (fun c => rightStatistic c.1) | |
| (sharingDesignated sharing designatedZLeft) | |
| (sharingDesignated sharing (designatedZRight parentShape)) leftLaw rightLaw χ 2 e z := by | |
| apply compatible_of_designated_windows | |
| · intro h u hn hd | |
| rw [hlaw h u hn hd.1 hd.2] | |
| have hp := (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft | |
| hsupport e x y z hsource hx hy hz h u hn hd.1).1 | |
| by_cases hzeroY : shapeNat u 1 = 0 | |
| · rw [ite_eq_left hzeroY] | |
| exact zeroY_transfer (fun _ : Class counts e h u => shapeNat u) | |
| (fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1) hp | |
| (leftStatistic h) (κLeft h) (hstatLeft h) (xLeftLaw (h, u)) χ | |
| (fun _ => hzeroY) (huseX h u hn hd.1).1 | |
| · rw [ite_eq_right hzeroY] | |
| have hzeroX : shapeNat u 0 = 0 := | |
| (Nat.mul_eq_zero.mp hd.2).resolve_right hzeroY | |
| exact zeroX_transfer (fun _ : Class counts e h u => shapeNat u) | |
| (fun i => (x h i.val).1) (fun i => (y h i.val).1) (fun i => (z h i.val).1) hp | |
| (leftStatistic h) (κLeft h) (hstatLeft h) (yLeftLaw (h, u)) χ | |
| (fun _ => hzeroX) (huseY h u hn hd.1).1 | |
| · intro h u hn hd | |
| rw [hrlaw h u hn hd.1 hd.2] | |
| have hp := (class_half_products_nonzero counts leftLength rightLength parentShape sharing hleft | |
| hsupport e x y z hsource hx hy hz h u hn hd.1).2 | |
| by_cases hzeroY : (parentShape h - shapeNat u) 1 = 0 | |
| · rw [ite_eq_left hzeroY] | |
| exact zeroY_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u) | |
| (fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2) hp | |
| (rightStatistic h) (κRight h) (hstatRight h) (xRightLaw (h, u)) χ | |
| (fun _ => hzeroY) (huseX h u hn hd.1).2 | |
| · rw [ite_eq_right hzeroY] | |
| have hzeroX : (parentShape h - shapeNat u) 0 = 0 := | |
| (Nat.mul_eq_zero.mp hd.2).resolve_right hzeroY | |
| exact zeroX_transfer (fun _ : Class counts e h u => parentShape h - shapeNat u) | |
| (fun i => (x h i.val).2) (fun i => (y h i.val).2) (fun i => (z h i.val).2) hp | |
| (rightStatistic h) (κRight h) (hstatRight h) (yRightLaw (h, u)) χ | |
| (fun _ => hzeroX) (huseY h u hn hd.1).2 | |
| end MatrixMultiplication.JointPopulationMixedCompatibility | |
| end | |
| end OAI | |