simp [Complex.toPolar, Polar.toComplex]
case mk
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag }
case mk
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle ∧
    Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag