Interaction Trees

A Library for Representing Recursive and Impure Programs in Coq

Introduction

For a quick overview of the core features of the library, see examples/ReadmeExample.v.

Top-level modules

ITree.ITree: Definitions to program with interaction trees.
ITree.ITreeFacts: Theorems to reason about interaction trees.
ITree.Events: Some standard event types.

Installation

Via opam

opam install coq-itree

Dependencies

See coq-itree.opam for version details.

Axioms

This library currently depends on UIP, functional extensionality, excluded middle, and choice; see also theories/Axioms.v.

UIP

This library depends on UIP for the inversion lemma:

Lemma eqit_inv_Vis
  : eutt eq (Vis e k1) (Vis e k2) ->
    forall x, eutt eq (k1 x) (k2 x).

There are a few more lemmas that depend on it, but you might not actually need it. For example, the compiler proof in tutorial doesn't need it and is axiom-free.

That lemma also has a weaker, but axiom-free version using heterogeneous equality: eqit_inv_Vis_weak.

The axiom that's technically used here is eq_rect_eq (and also JMeq_eq in old versions of Coq), which is equivalent to UIP.

Functional extensionality

The closed category of functions assumes functional_extensionality, in Basics.FunctionFacts.CartesianClosed_Fun.

Excluded middle and choice

The theory of traces (theories/ITrace/)—which Dijkstra monads for ITree depend on (theories/Dijkstra)—assumes excluded middle, to decide whether an itree diverges, and a type-theoretic axiom of choice, which provides a strong excluded middle in propositional contexts:

Theorem classicT_inhabited : inhabited (forall T : Type, T + (T -> False)).

Remark: excluded middle implies UIP, but we still consider UIP as a separate dependency because it's used in a more central part of the library.

Exported: strong bisimulation is propositional equality

The library exports the following axiom for convenience, though it's unlikely you'll need it, and the rest of the library does not depend on it:

Axiom bisimulation_is_eq : t1 ≅ t2 -> t1 = t2.

Contributions welcome

Feel free to open an issue or a pull request!

See also DEV.md for working on this library.

DeepSpec/InteractionTrees