openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Tensor /ComplexInitialTargetKernel.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialControlledKernel | |
| namespace OAI | |
| /-! Finite coefficient tensors and their algebraic transformations. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.InitialTargetKernel | |
| open Tensor LocalMaps PolynomialLocalConstruction | |
| variable {X Y Z RawLabel Label : Type*} | |
| def Chosen (window : X → Prop) (read : X → RawLabel) (embed : Label → RawLabel) | |
| (label : Label) (x : X) : Prop := | |
| window x ∧ read x = embed label | |
| def Eligible (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) (label : Label) : X → Y → Z → Prop := | |
| fun x y z => Chosen PX lx embed label x ∧ Chosen PY ly embed label y ∧ | |
| Chosen PZ lz embed label z | |
| def auxiliary : Tensor ℂ Unit Unit Unit := fun _ _ _ => 1 | |
| theorem auxiliary_rank : RankAtMost auxiliary 1 := by | |
| have haux : auxiliary = rankOne (fun _ : Unit => (1 : ℂ)) | |
| (fun _ : Unit => (1 : ℂ)) (fun _ : Unit => (1 : ℂ)) := by | |
| funext x y z | |
| simp only [auxiliary, rankOne, one_mul] | |
| rw [haux] | |
| exact rankOne_rankAtMost _ _ _ | |
| def sideMap (window : X → Prop) (read : X → RawLabel) (embed : Label → RawLabel) : | |
| X → (Label × (Fin 1 × Fin 1)) → Unit → Polynomial ℂ := by | |
| classical | |
| exact fun x output _ => Polynomial.C (if Chosen window read embed output.1 x then 1 else 0) | |
| theorem sideMap_degree (window : X → Prop) (read : X → RawLabel) | |
| (embed : Label → RawLabel) (x : X) (output : Label × (Fin 1 × Fin 1)) (input : Unit) : | |
| (sideMap window read embed x output input).degree ≤ (0 : ℕ) := by | |
| classical | |
| exact Polynomial.degree_C_le | |
| def rawLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) : | |
| Tensor ℂ (X × (Label × (Fin 1 × Fin 1))) | |
| (Y × (Label × (Fin 1 × Fin 1))) (Z × (Label × (Fin 1 × Fin 1))) := by | |
| classical | |
| exact fun x y z => if Chosen PX lx embed x.2.1 x.1 ∧ | |
| Chosen PY ly embed y.2.1 y.1 ∧ Chosen PZ lz embed z.2.1 z.1 then 1 else 0 | |
| def previousLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) : | |
| Tensor ℂ (X × (Label × (Fin 1 × Fin 1))) | |
| (Y × (Label × (Fin 1 × Fin 1))) (Z × (Label × (Fin 1 × Fin 1))) := by | |
| classical | |
| exact fun x y z => if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 then 1 else 0 | |
| theorem kernel_eq_C_rawLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) | |
| (x : X × (Label × (Fin 1 × Fin 1))) | |
| (y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) : | |
| kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed) | |
| x y z = Polynomial.C (rawLeading PX PY PZ lx ly lz embed x y z) := by | |
| classical | |
| by_cases hx : Chosen PX lx embed x.2.1 x.1 <;> | |
| by_cases hy : Chosen PY ly embed y.2.1 y.1 <;> | |
| by_cases hz : Chosen PZ lz embed z.2.1 z.1 <;> | |
| simp [kernel, fiberTransform, sideMap, auxiliary, rawLeading, hx, hy, hz] | |
| theorem rawLeading_eq_previousLeading | |
| (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) (injective : Function.Injective embed) | |
| (support : X → Y → Z → Prop) | |
| (compatible : ∀ x y z, support x y z → PX x → PY y → PZ z → | |
| lx x = ly y ∧ lx x = lz z) | |
| (x : X × (Label × (Fin 1 × Fin 1))) | |
| (y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) | |
| (hs : support x.1 y.1 z.1) : | |
| rawLeading PX PY PZ lx ly lz embed x y z = | |
| previousLeading PX PY PZ lx ly lz embed x y z := by | |
| classical | |
| have hcondition : | |
| (Chosen PX lx embed x.2.1 x.1 ∧ Chosen PY ly embed y.2.1 y.1 ∧ | |
| Chosen PZ lz embed z.2.1 z.1) ↔ | |
| (x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1) := by | |
| constructor | |
| · rintro ⟨hx, hy, hz⟩ | |
| have hsync := compatible x.1 y.1 z.1 hs hx.1 hy.1 hz.1 | |
| have hxy : x.2.1 = y.2.1 := injective (hx.2.symm.trans (hsync.1.trans hy.2)) | |
| have hxz : x.2.1 = z.2.1 := injective (hx.2.symm.trans (hsync.2.trans hz.2)) | |
| refine ⟨hxy, hxz, hx, ?_, ?_⟩ | |
| · simpa only [hxy] using hy | |
| · simpa only [hxz] using hz | |
| · rintro ⟨hxy, hxz, hx, hy, hz⟩ | |
| refine ⟨hx, ?_, ?_⟩ | |
| · simpa only [hxy] using hy | |
| · simpa only [hxz] using hz | |
| simp only [rawLeading, previousLeading, hcondition] | |
| theorem kernel_leading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) (injective : Function.Injective embed) | |
| (support : X → Y → Z → Prop) | |
| (compatible : ∀ x y z, support x y z → PX x → PY y → PZ z → | |
| lx x = ly y ∧ lx x = lz z) | |
| (x : X × (Label × (Fin 1 × Fin 1))) | |
| (y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) | |
| (hs : support x.1 y.1 z.1) : | |
| (kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed) | |
| x y z).coeff 0 = previousLeading PX PY PZ lx ly lz embed x y z := by | |
| rw [kernel_eq_C_rawLeading, Polynomial.coeff_C_zero] | |
| exact rawLeading_eq_previousLeading PX PY PZ lx ly lz embed injective support | |
| compatible x y z hs | |
| theorem kernel_low (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) | |
| (x : X × (Label × (Fin 1 × Fin 1))) | |
| (y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) | |
| (j : ℕ) (hj : j < 0) : | |
| (kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed) | |
| x y z).coeff j = 0 := | |
| (Nat.not_lt_zero j hj).elim | |
| theorem previousLeading_ne_zero_iff (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) | |
| (x : X × (Label × (Fin 1 × Fin 1))) | |
| (y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) : | |
| previousLeading PX PY PZ lx ly lz embed x y z ≠ 0 ↔ | |
| x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 := by | |
| classical | |
| by_cases h : x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 <;> | |
| simp [previousLeading, h] | |
| theorem kernel_substitution (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop) | |
| (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel) | |
| (embed : Label → RawLabel) (original : Tensor ℂ X Y Z) : | |
| fiberTransform (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed) | |
| (fun x y z => Polynomial.C (Tensor.product original auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)) := by | |
| have h := fiberTransform_originalScale (sideMap PX lx embed) | |
| (sideMap PY ly embed) (sideMap PZ lz embed) | |
| (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (auxiliary x.2 y.2 z.2)) | |
| have hinput : | |
| (fun x y z => Polynomial.C (Tensor.product original auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (auxiliary x.2 y.2 z.2)) := by | |
| funext x y z | |
| simp only [Tensor.product, originalScale, map_mul] | |
| rw [hinput] | |
| simpa only [kernel] using h | |
| end MatrixMultiplication.Foundation.InitialTargetKernel | |
| end | |
| end OAI | |