Section: New Results

Semantics of the Calculus of Inductive Constructions

Participant : Bruno Barras [Contact] .

Bruno Barras has formalized the meta-theoretical study of strictly positive inductive types. This was built upon the previous work on specific instances: natural numbers and Brouwer ordinals. The main idea of the model construction is to use the property that every strictly positive inductive definition can be encoded in the parameterized type of trees (the so-called W-types). Such tree-types can themselves be encoded as partial functions from paths to labels. The soundness of this translation gives a way to build the closure ordinal of any strictly positive inductive definition.

Bruno Barras has then modelled the inductive families (also called inductive types with indices). He has been able to prove formally the previously known result that inductive family can be constructed in two steps: first build a carrier type (inductively) which is oblivious of the indices, and then define each member of the family as a subset of the carrier type by enforcing the constraints generated by the indices.

He also started to investigate advanced features of inductive definitions, like the possibility to have non-uniform parameters. When this feature was introduced in Coq , it was thought as a conservative one, but the formal analysis showed that this was not obvious. The consistency model could be extended (with one auxiliary result not yet encoded formally). This shows that non-uniform parameters do not extend much the expressivity of Coq , but the strict equivalence remained as an open problem.