openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileStateInterpretation.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileTaylorStep | |
| namespace OAI | |
| /-! Interpretation of every field in the exact profile-certificate recurrence. -/ | |
| open Matrix | |
| namespace DefocusingNLS.ProfileCertificate | |
| @[simp] theorem complexMatrix_add (A B : QMatrix) : | |
| complexMatrix (Matrix2.add A B) = complexMatrix A+complexMatrix B := by | |
| ext i j | |
| simp only [complexMatrix, Matrix2.toMatrix_add, Matrix.map_apply, Matrix.add_apply, map_add] | |
| @[simp] theorem rationalMatrix_add (E F : RMatrix) : | |
| rationalMatrix (Matrix2.add E F) = rationalMatrix E+rationalMatrix F := by | |
| ext i j | |
| simp only [rationalMatrix, Matrix2.toMatrix_add, Matrix.map_apply, Matrix.add_apply, map_add] | |
| @[simp] theorem rationalMatrix_mul (E F : RMatrix) : | |
| rationalMatrix (Matrix2.mul E F) = rationalMatrix E*rationalMatrix F := by | |
| simp only [rationalMatrix, Matrix2.toMatrix_mul, Matrix.map_mul] | |
| @[simp] theorem rationalMatrix_scale (r : ℚ) (E : RMatrix) : | |
| rationalMatrix (Matrix2.scale r E) = (r : ℝ) • rationalMatrix E := by | |
| simp only [rationalMatrix, Matrix2.toMatrix_scale] | |
| exact Matrix.map_smul' _ _ _ (Rat.castHom ℝ).map_mul | |
| @[simp] theorem complexMatrix_scale_real (r : ℚ) (A : QMatrix) : | |
| complexMatrix (Matrix2.scale (⟨r,0⟩ : RationalComplex) A) = (r : ℝ) • complexMatrix A := by | |
| rw [complexMatrix_scale] | |
| have he : RationalComplex.toComplex (⟨r,0⟩ : RationalComplex) = ((r : ℝ) : ℂ) := by | |
| apply Complex.ext <;> simp [RationalComplex.toComplex] | |
| rw [he] | |
| ext i j | |
| exact (Complex.real_smul).symm | |
| theorem state_value_step (n : ℕ) (s : State) : | |
| complexMatrix (step n s).value = ((n : ℝ)+1)⁻¹ • | |
| (complexMatrix (factor n)*complexMatrix s.value) := by | |
| simp only [step, complexMatrix_scale_real, complexMatrix_mul] | |
| simp only [Rat.cast_inv, Rat.cast_add, Rat.cast_natCast, Rat.cast_one] | |
| theorem state_derivB_step (n : ℕ) (s : State) : | |
| complexMatrix (step n s).derivB = ((n : ℝ)+1)⁻¹ • | |
| (complexMatrix parameterB*complexMatrix s.value+ | |
| complexMatrix (factor n)*complexMatrix s.derivB) := by | |
| simp only [step, complexMatrix_scale_real, complexMatrix_add, complexMatrix_mul] | |
| simp only [Rat.cast_inv, Rat.cast_add, Rat.cast_natCast, Rat.cast_one] | |
| theorem state_derivZ_step (n : ℕ) (s : State) : | |
| complexMatrix (step n s).derivZ = ((n : ℝ)+1)⁻¹ • | |
| (complexMatrix parameterZ*complexMatrix s.value+ | |
| complexMatrix (factor n)*complexMatrix s.derivZ) := by | |
| simp only [step, complexMatrix_scale_real, complexMatrix_add, complexMatrix_mul] | |
| simp only [Rat.cast_inv, Rat.cast_add, Rat.cast_natCast, Rat.cast_one] | |
| theorem state_error_step (n : ℕ) (s : State) : | |
| rationalMatrix (step n s).error = ((n : ℝ)+1)⁻¹ • | |
| ((absoluteMatrix (factor n)+(radius : ℝ) • | |
| (absoluteMatrix parameterB+absoluteMatrix parameterZ))*rationalMatrix s.error+ | |
| (absoluteMatrix parameterB+absoluteMatrix parameterZ)* | |
| (absoluteMatrix s.derivB+absoluteMatrix s.derivZ)) := by | |
| simp only [step, rationalMatrix_scale, rationalMatrix_add, rationalMatrix_mul, | |
| majorant, absoluteMatrix] | |
| simp only [Rat.cast_inv, Rat.cast_add, Rat.cast_natCast, Rat.cast_one] | |
| theorem profileFactor_affine (n : ℕ) (b z : ℝ) : | |
| profileFactor ((centerB : ℝ)+b) ((centerZ : ℝ)+z) n = ((n : ℝ)+1)⁻¹ • | |
| (complexMatrix (factor n)+b • complexMatrix parameterB+z • complexMatrix parameterZ) := by | |
| ext i j | |
| fin_cases i <;> fin_cases j <;> | |
| simp [profileFactor, complexMatrix, factor, parameterB, parameterZ, | |
| Matrix2.toMatrix, backwardMatrix, RationalComplex.toComplex, Complex.real_smul, | |
| Complex.ext_iff] <;> ring | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |