Univalent Foundations of Constructive Algebraic Geometry

We investigate two constructive approaches to defining quasi-compact and quasi-separated schemes (qcqs-schemes), namely qcqs-schemes as locally ringed lattices and as functors from rings to sets. We work in Homotopy Type Theory and Univalent Foundations, but reason informally. The main result is a constructive and univalent proof that the two definitions coincide, giving an equivalence between the respective categories of qcqs-schemes.