openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionDrop /DimensionDropEndpointGenerators.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropFibres | |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropFibreGenerators | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped ComplexOrder CStarAlgebra | |
| namespace CurrentMain.ProjectionRow | |
| open ConcreteMatrixUnits | |
| variable {I B : Type*} [Fintype I] [DecidableEq I] [CStarAlgebra B] | |
| [PartialOrder B] [StarOrderedRing B] {q : B} | |
| omit [PartialOrder B] [StarOrderedRing B] in | |
| lemma matrixHom_unit (V : ProjectionRow I q) (i j : I) : | |
| V.matrixHom (unit i j)=V.entry i*star (V.entry j) := by | |
| change (∑ k : I,∑ l : I,(unit i j k l) • V.coeff q k l)=_ | |
| simp only [unit,CStarMatrix.ofMatrix_apply,Matrix.single_apply,ite_and,ite_smul,one_smul,zero_smul] | |
| simp only [Finset.sum_ite_irrel,Finset.sum_const_zero,Finset.sum_ite_eq,Finset.mem_univ,ite_true] | |
| rw [coeff,V.right_support] | |
| end CurrentMain.ProjectionRow | |
| namespace DimensionDropRelations.Witness | |
| open ConcreteMatrixUnits CurrentMain CompletelyPositiveOperations | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| variable {B : Type*} [CStarAlgebra B] [PartialOrder B] [StarOrderedRing B] | |
| variable {p : I} (w : Witness (B := B) p) | |
| lemma lowerEndpointHom_phi (hc : w.parameter=0) (i j : I) : | |
| w.lowerEndpointHom hc (unit (some i) (some j))=w.φ (matrixUnit i j) := by | |
| change (w.fibreRow (t := 0) (by simpa using hc) (by norm_num)).matrixHom (unit (some i) (some j))=_ | |
| rw [ProjectionRow.matrixHom_unit] | |
| simpa only [fibreRow,fibreEntry,fibreColumn,sub_zero,Real.sqrt_one,inv_one,one_smul] using | |
| (w.zero_fibre_phi_matrix hc i j).symm | |
| lemma lowerEndpointHom_s (hc : w.parameter=0) : | |
| w.lowerEndpointHom hc (unit (some p) none)=w.s := by | |
| let V := w.fibreRow (t := 0) (by simpa using hc) (by norm_num) | |
| change V.matrixHom (unit (some p) none)=w.s | |
| rw [ProjectionRow.matrixHom_unit] | |
| change V.entry (some p)*star (w.fibreAnchor 0)=w.s | |
| rw [w.fibreAnchor_star] | |
| rw [V.right_support] | |
| change (Real.sqrt (1-(0:ℝ)))⁻¹ • w.column p=w.s | |
| simpa only [sub_zero,Real.sqrt_one,inv_one,one_smul,column] using w.support | |
| lemma upperEndpointHom_phi (hc : w.parameter=1) (i j : I) : | |
| w.upperEndpointHom hc (unit i j)=w.φ (matrixUnit i j) := rfl | |
| end DimensionDropRelations.Witness | |
| end | |
| end | |
| end OAI | |