We propose a logic for temporal properties of higher-order programs that handle infinite objects like streams or infinite trees, rep- resented via coinductive types. Specifications of programs use safety and liveness properties. Programs can then be proven to satisfy their specifi- cation in a compositional way, our logic being based on a type system. The logic is presented as a refinement type system over the guarded λ-calculus, a λ-calculus with guarded recursive types. The refinements are formulae of a modal μ-calculus which embeds usual temporal modal logics such as LTL and CTL. The semantics of our system is given within a rich structure, the topos of trees, in which we build a realizability model of the temporal refinement type system. We use in a crucial way the connection with set-theoretic semantics to handle liveness properties.

Temporal Refinements for Guarded Recursive Types

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