Invariants for the FoCaL language. We present a FoCaL formalization for quotient structures which are common in mathematics. We first present a framework for stating invariant properties of the data manipulated by running programs. A notion of equivalence relation is then encoded for the FoCaL library. It is implemented through projections functions, this enables us to provide canonical representations which are commonly used in Computer Algebra but seldom formally described. We further provide a FoCaL formalization for the code used inside the library for modular arithmetic through the certification of quotient groups and quotient rings which are involved in the model. We finally instantiate our framework to provide a trusted replacement of the existing FoCaL library.
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