Interpretation of Inaccessible Sets in Martin-Löf Type Theory with One Mahlo Universe

Martin-L\"{o}f type theory $\mathbf{MLTT}$ was extended by Setzer with the so-called Mahlo universe types. The extension of $\mathbf{MLTT}$ with one Mahlo universe is called $\mathbf{MLM}$ and was introduced to develop a variant of $\mathbf{MLTT}$ equipped with an analogue of a large cardinal. Another instance of constructive systems extended with an analogue of a large set was formulated in the context of Aczel's constructive set theory: $\mathbf{CZF}$. Rathjen, Griffor and Palmgren extended $\mathbf{CZF}$ with inaccessible sets of all transfinite orders. While Rathjen proved that this extended system of $\mathbf{CZF}$ is interpretable in an extension of $\mathbf{MLM}$ with one usual universe type above the Mahlo universe, it is unknown whether it can be interpreted by the Mahlo universe without a universe type above it. We extend $\mathbf{MLM}$ not by a universe type but by the accessibility predicate, and show that $\mathbf{CZF}$ with inaccessible sets can be interpreted in $\mathbf{MLM}$ with the accessibility predicate. Our interpretation of this extension of $\mathbf{CZF}$ is the same as that of Rathjen, Griffor and Palmgren formulated by $\mathbf{MLTT}$ with second-order universe operators, except that we construct the inaccessible sets by using the Mahlo universe and the accessibility predicate. We formalised the main part of our interpretation in the proof assistant Agda.