openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldHistory /ProducedBase.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Positions | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalCW | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FiniteFamily | |
| namespace OAI | |
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldHistory | |
| open MatrixMultiplication.Foundation AllFieldParameters JointPopulation | |
| open CWStrands HistorySymmetry InheritedMasks JointCanonicalization | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| abbrev CanonicalWords := CanonicalPairs (activeCounts (K := K) (tick := tick) allocation dilation) | |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) | |
| abbrev ProducedWords := ∀ h : Produced K tick, | |
| Fin (population allocation dilation h.val) → Fin (currentLength h.val.1) → Fin 7 | |
| abbrev CanonicalClassPosition := | |
| Σ c : Active K tick × JointPopulation.Shape, Fin (activeCounts allocation dilation c.1 c.2) | |
| abbrev CanonicalHalfPosition := | |
| CanonicalClassPosition (K := K) (tick := tick) allocation dilation × Bool | |
| def canonicalClassEquiv : CanonicalClassPosition (K := K) (tick := tick) allocation dilation ≃ | |
| (Σ h : Active K tick, Σ u : JointPopulation.Shape, Fin (activeCounts allocation dilation h u)) where | |
| toFun p := ⟨p.1.1, p.1.2, p.2⟩ | |
| invFun p := ⟨(p.1, p.2.1), p.2.2⟩ | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| def splitArrangeEquiv : | |
| (Σ h : Active K tick, Σ b : h.val.1.Branch, | |
| Fin (branchPopulation allocation dilation h.val b)) × Bool ≃ | |
| SplitPositions (K := K) (tick := tick) allocation dilation where | |
| toFun p := ⟨p.1.1, p.1.2.1, p.2, p.1.2.2⟩ | |
| invFun p := (⟨p.1, p.2.1, p.2.2.2⟩, p.2.2.1) | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| def canonicalSplitEquiv : | |
| CanonicalHalfPosition (K := K) (tick := tick) allocation dilation ≃ | |
| SplitPositions (K := K) (tick := tick) allocation dilation := | |
| (Equiv.prodCongr | |
| ((canonicalClassEquiv allocation dilation).trans | |
| (Equiv.sigmaCongrRight fun h : Active K tick => | |
| branchPositionEquiv allocation dilation h.val)) | |
| (Equiv.refl Bool)).trans (splitArrangeEquiv allocation dilation) | |
| def canonicalProducedEquiv : | |
| CanonicalHalfPosition (K := K) (tick := tick) allocation dilation ≃ | |
| ProducedPositions (K := K) (tick := tick) allocation dilation := | |
| (canonicalSplitEquiv allocation dilation).trans | |
| (producedPositionEquiv allocation dilation).symm | |
| theorem splitProduced_length (p : SplitPositions (K := K) (tick := tick) allocation dilation) : | |
| currentLength ((producedPositionEquiv allocation dilation).symm p).1.val.1 = | |
| activeHalfLength p.1 := by | |
| simpa only [Equiv.apply_symm_apply, activeHalfLength] using | |
| producedPositionEquiv_length allocation dilation | |
| ((producedPositionEquiv allocation dilation).symm p) | |
| def splitWord (x : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (p : SplitPositions (K := K) (tick := tick) allocation dilation) : | |
| Fin (activeHalfLength p.1) → Fin 7 := | |
| let q := (producedPositionEquiv allocation dilation).symm p | |
| fun i => x q.1 q.2 (Fin.cast (splitProduced_length allocation dilation p).symm i) | |
| def producedHalfWord (x : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (h : Active K tick) (b : h.val.1.Branch) (right : Bool) : | |
| Fin (branchPopulation allocation dilation h.val b) → Fin (activeHalfLength h) → Fin 7 := | |
| fun i => splitWord allocation dilation x ⟨h, b, right, i⟩ | |
| def regroupAt (x : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| Fin (activeHalfLength p.1.1.1) → Fin 7 := | |
| splitWord allocation dilation x (canonicalSplitEquiv allocation dilation p) | |
| def regroupWords (x : ProducedWords (K := K) (tick := tick) allocation dilation) : | |
| CanonicalWords (K := K) (tick := tick) allocation dilation where | |
| left c i := regroupAt allocation dilation x (⟨c, i⟩, false) | |
| right c i := regroupAt allocation dilation x (⟨c, i⟩, true) | |
| def branchClassEquiv (h : Active K tick) (b : h.val.1.Branch) : | |
| Fin (branchPopulation allocation dilation h.val b) ≃ | |
| Fin (activeCounts allocation dilation h (branchShape h.val b)) := | |
| { toFun := fun i => ⟨i.val, by | |
| change i.val < jointCounts allocation dilation h.val (branchShape h.val b) | |
| rw [jointCounts_at] | |
| exact i.isLt⟩ | |
| invFun := fun i => ⟨i.val, by | |
| simpa only [activeCounts, jointCounts_at] using i.isLt⟩ | |
| left_inv := fun _ => rfl | |
| right_inv := fun _ => rfl } | |
| def branchClassIndex (h : Active K tick) (b : h.val.1.Branch) | |
| (i : Fin (branchPopulation allocation dilation h.val b)) : | |
| Fin (activeCounts allocation dilation h (branchShape h.val b)) := | |
| branchClassEquiv allocation dilation h b i | |
| theorem regroupWords_branch (x : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (h : Active K tick) (b : h.val.1.Branch) (right : Bool) | |
| (i : Fin (branchPopulation allocation dilation h.val b)) : | |
| (if right then (regroupWords allocation dilation x).right | |
| (h, branchShape h.val b) (branchClassIndex allocation dilation h b i) | |
| else (regroupWords allocation dilation x).left | |
| (h, branchShape h.val b) (branchClassIndex allocation dilation h b i)) = | |
| producedHalfWord allocation dilation x h b right i := by | |
| have he := (branchPositionEquiv allocation dilation h.val).apply_symm_apply ⟨b, i⟩ | |
| cases right <;> simp only [Bool.false_eq_true, ↓reduceIte] | |
| · exact congrArg (fun q : Σ b : h.val.1.Branch, | |
| Fin (branchPopulation allocation dilation h.val b) => | |
| producedHalfWord allocation dilation x h q.1 false q.2) he | |
| · exact congrArg (fun q : Σ b : h.val.1.Branch, | |
| Fin (branchPopulation allocation dilation h.val b) => | |
| producedHalfWord allocation dilation x h q.1 true q.2) he | |
| def canonicalHalfShape | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : AllFieldParameters.Shape := | |
| if p.2 then activeParentShape p.1.1.1 - JointCanonicalCW.shapeNat p.1.1.2 | |
| else JointCanonicalCW.shapeNat p.1.1.2 | |
| theorem canonicalProduced_length | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| currentLength (canonicalProducedEquiv allocation dilation p).1.val.1 = | |
| activeHalfLength p.1.1.1 := | |
| splitProduced_length allocation dilation (canonicalSplitEquiv allocation dilation p) | |
| theorem canonicalProduced_shape | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| currentPhysicalShape (canonicalProducedEquiv allocation dilation p).1.val = | |
| canonicalHalfShape allocation dilation p := by | |
| let w := p.1.1.1 | |
| let q := branchPositionEquiv allocation dilation w.val ⟨p.1.1.2, p.1.2⟩ | |
| have hu : physicalShape w.val.2 (w.val.1.splitShape q.1) = | |
| JointCanonicalCW.shapeNat p.1.1.2 := by | |
| have hh := congrArg decodeShape | |
| (branchPositionEquiv_shape allocation dilation w.val ⟨p.1.1.2, p.1.2⟩) | |
| simp only [branchShape, decode_encodePhysicalShape] at hh | |
| exact hh | |
| let shapeOfSplit (r : SplitPositions (K := K) (tick := tick) allocation dilation) : | |
| AllFieldParameters.Shape := | |
| physicalShape r.1.val.2 (halfShape r.1.val.1.parentShape | |
| (r.1.val.1.splitShape r.2.1) r.2.2.1) | |
| have hs : currentPhysicalShape | |
| ((producedPositionEquiv allocation dilation).symm | |
| (canonicalSplitEquiv allocation dilation p)).1.val = | |
| shapeOfSplit ((producedPositionEquiv allocation dilation) | |
| ((producedPositionEquiv allocation dilation).symm | |
| (canonicalSplitEquiv allocation dilation p))) := | |
| producedPositionEquiv_shape allocation dilation | |
| ((producedPositionEquiv allocation dilation).symm | |
| (canonicalSplitEquiv allocation dilation p)) | |
| rw [Equiv.apply_symm_apply] at hs | |
| change currentPhysicalShape (canonicalProducedEquiv allocation dilation p).1.val = | |
| physicalShape w.val.2 (halfShape w.val.1.parentShape | |
| (w.val.1.splitShape q.1) p.2) at hs | |
| rw [hs] | |
| cases hr : p.2 <;> | |
| simp [canonicalHalfShape, halfShape, hr, physicalShape_complement, hu, | |
| activeParentShape] | |
| rfl | |
| variable {F : Type*} [Field F] | |
| theorem shapeTensor_finCast {m n : ℕ} (h : m = n) (g : AllFieldParameters.Shape) | |
| (x y z : Fin n → Fin 7) : | |
| shapeTensor (F := F) (Fin m) g (x ∘ Fin.cast h) (y ∘ Fin.cast h) (z ∘ Fin.cast h) = | |
| shapeTensor (Fin n) g x y z := by | |
| cases h | |
| rfl | |
| def canonicalChildren : Tensor F | |
| (CanonicalWords (K := K) (tick := tick) allocation dilation) | |
| (CanonicalWords (K := K) (tick := tick) allocation dilation) | |
| (CanonicalWords (K := K) (tick := tick) allocation dilation) := | |
| pairClassProduct (P := ClassPositions (activeCounts allocation dilation)) | |
| (JointCanonicalCW.leftTensor activeHalfLength) | |
| (JointCanonicalCW.rightTensor activeHalfLength activeParentShape) | |
| def producedBase : Tensor F | |
| (ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (ProducedWords (K := K) (tick := tick) allocation dilation) := | |
| CommonDimensions.familyProduct (fun h : Produced K tick => | |
| Tensor.power (shapeTensor (Fin (currentLength h.val.1)) (currentPhysicalShape h.val)) | |
| (population allocation dilation h.val)) | |
| def canonicalHalfWord (x : CanonicalWords (K := K) (tick := tick) allocation dilation) | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| Fin (activeHalfLength p.1.1.1) → Fin 7 := | |
| if p.2 then x.right p.1.1 p.1.2 else x.left p.1.1 p.1.2 | |
| theorem regroupWords_half (x : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| canonicalHalfWord allocation dilation (regroupWords allocation dilation x) p = | |
| regroupAt allocation dilation x p := by | |
| rcases p with ⟨⟨c, i⟩, right⟩ | |
| cases right <;> simp only [canonicalHalfWord, regroupWords, Bool.false_eq_true, ↓reduceIte] | |
| theorem canonicalChildren_flat | |
| (x y z : CanonicalWords (K := K) (tick := tick) allocation dilation) : | |
| canonicalChildren (F := F) allocation dilation x y z = | |
| ∏ p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation, | |
| shapeTensor (Fin (activeHalfLength p.1.1.1)) (canonicalHalfShape allocation dilation p) | |
| (canonicalHalfWord allocation dilation x p) | |
| (canonicalHalfWord allocation dilation y p) | |
| (canonicalHalfWord allocation dilation z p) := by | |
| simp only [CanonicalHalfPosition, Fintype.prod_prod_type, CanonicalClassPosition, | |
| Fintype.prod_sigma, Fintype.prod_bool, canonicalHalfShape, canonicalHalfWord, | |
| Bool.false_eq_true, ↓reduceIte, Finset.prod_mul_distrib, | |
| canonicalChildren, pairClassProduct, classProduct, JointCanonicalCW.leftTensor, | |
| JointCanonicalCW.rightTensor] | |
| exact mul_comm _ _ | |
| theorem producedBase_flat | |
| (x y z : ProducedWords (K := K) (tick := tick) allocation dilation) : | |
| producedBase (F := F) allocation dilation x y z = | |
| ∏ p : ProducedPositions (K := K) (tick := tick) allocation dilation, | |
| shapeTensor (Fin (currentLength p.1.val.1)) (currentPhysicalShape p.1.val) | |
| (x p.1 p.2) (y p.1 p.2) (z p.1 p.2) := by | |
| simp only [producedBase, CommonDimensions.familyProduct, Tensor.power, | |
| ProducedPositions, Fintype.prod_sigma] | |
| theorem regroup_coefficient_at | |
| (x y z : ProducedWords (K := K) (tick := tick) allocation dilation) | |
| (p : CanonicalHalfPosition (K := K) (tick := tick) allocation dilation) : | |
| shapeTensor (F := F) (Fin (activeHalfLength p.1.1.1)) (canonicalHalfShape allocation dilation p) | |
| (regroupAt allocation dilation x p) (regroupAt allocation dilation y p) | |
| (regroupAt allocation dilation z p) = | |
| shapeTensor (Fin (currentLength (canonicalProducedEquiv allocation dilation p).1.val.1)) | |
| (currentPhysicalShape (canonicalProducedEquiv allocation dilation p).1.val) | |
| (x (canonicalProducedEquiv allocation dilation p).1 (canonicalProducedEquiv allocation dilation p).2) | |
| (y (canonicalProducedEquiv allocation dilation p).1 (canonicalProducedEquiv allocation dilation p).2) | |
| (z (canonicalProducedEquiv allocation dilation p).1 (canonicalProducedEquiv allocation dilation p).2) := by | |
| rw [← canonicalProduced_shape allocation dilation p] | |
| exact shapeTensor_finCast (canonicalProduced_length allocation dilation p).symm _ _ _ _ | |
| theorem regroup_coefficient | |
| (x y z : ProducedWords (K := K) (tick := tick) allocation dilation) : | |
| canonicalChildren (F := F) allocation dilation | |
| (regroupWords allocation dilation x) (regroupWords allocation dilation y) | |
| (regroupWords allocation dilation z) = producedBase allocation dilation x y z := by | |
| rw [canonicalChildren_flat, producedBase_flat] | |
| simp only [regroupWords_half] | |
| exact Fintype.prod_equiv (canonicalProducedEquiv allocation dilation) _ _ | |
| (regroup_coefficient_at allocation dilation x y z) | |
| theorem regroup_pullback : | |
| Tensor.pullback (regroupWords allocation dilation) (regroupWords allocation dilation) | |
| (regroupWords allocation dilation) (canonicalChildren (F := F) allocation dilation) = | |
| producedBase (K := K) (tick := tick) allocation dilation := by | |
| funext x y z | |
| exact regroup_coefficient allocation dilation x y z | |
| def regroupMatrix : | |
| ProducedWords (K := K) (tick := tick) allocation dilation → | |
| CanonicalWords (K := K) (tick := tick) allocation dilation → F := | |
| fun x s => if s = regroupWords allocation dilation x then 1 else 0 | |
| def producedBaseMap : | |
| AllFieldFiniteFamily.LocalMap | |
| (canonicalChildren (F := F) (K := K) (tick := tick) allocation dilation) | |
| (producedBase (F := F) (K := K) (tick := tick) allocation dilation) where | |
| x := regroupMatrix allocation dilation | |
| y := regroupMatrix allocation dilation | |
| z := regroupMatrix allocation dilation | |
| coefficient := by | |
| unfold regroupMatrix | |
| rw [← Tensor.pullback_eq_restrict] | |
| exact regroup_pullback allocation dilation | |
| end MatrixMultiplication.AllFieldHistory | |
| end | |
| end OAI | |