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proof-relevant-pi, release 0.3

Agda development accompanying the paper [Proof-relevant pi calculus] (, presented at LFMTP 2015, and also the foundation for the concurrent-slicing formalisation of dynamic slicing for the pi-calculus. See the appendix of the accompanying paper for notes on module structure.

Required compiler and libraries

  • Agda, version
  • Agda standard library, version 0.9.
  • agda-stdlib-ext library, version 0.0.3


  • Reorganise packages relating to cofinality and causal equivalence
  • Proof of symmetry for causal equivalence
  • Vertical composition of braidings
  • Several examples of causal equivalence (trickier than it should be)
  • Proof of ⊖-preserves-sym
  • Tighten connection to paper, e.g. key theorems, naming conventions
  • Typeclass for residuals (improve overloading mess somehow, and problem with use of Δ as type, constructor and meta-variable)

Design notes

0.3 CONCUR 2016 version

The "cofinality" relation currently relates two processes which are in the same context (as does the definition of the symmetric residual). This requires one of the two interleavings to determine the target context, with the implied coercion on the other process. A more symmetric version (based on the symmetry of ᴬ⌣) simplifies in some ways (some of these coercions disappear), but complexifies in others, since Agda does not know that symmetry is involutive. Decided to leave things as they are.

0.2 Precise cofinality

Implemented a more specific braiding relation, as described in my PLInG talk. Decided to drop the cube lemma and related formalisation (residuals of two-dimensional transitions), because the current approach to dealing with concurrent extrusions will require some rework to generalise to the cube setting. Suppose I have a 2D action representing a pair of extrusion-rendezvous, which is concurrent with a bound output. If the bound output is an extrusion of the same binder as one of the extrusion-rendezvous, the residual of the 2D action after the bound output will not be a pair of extrusion-rendezvous; one or both of the extrusion-rendezvous will have become a regular rendezvous. For 1D actions, where extrusion-rendezvous and regular rendezvous are not distinguished, this is covered by the rule that says that the residual of a non-bound action after a bound action is non-bound. The precise braiding approach requires a 2D action that distinguishes between extrusion-rendezvous and regular rendezvous, but without further representation of the subtleties of extrusion, there is no longer a functional relationship between the residual of a 2D action after a 1D action. Without such a function, it is not easy to define (as a function) the residual of a 2D transition after a 1D transition, since the function is used to give its type.

Since the cube lemma is a bit of a struggle with the whole braiding business (since 0.1 release notes below), it seems more sensible to drop that and retain the more precise cofinality formulation, which I think fits better with the rest of the paper. The cube lemma isn't required to define causal equivalence.

0.1 LFMTP final version

Wasted a solid week trying to prove a version of /-preserves-⌣ for a symmetric version of the relation. If I redefine ⌣ so it is a congruence by definition (with a symmetric variant of each rule), then the proof (essentially the definition of a two-dimensional residual) becomes enormous (many thousands of LOC) and Agda runs out of memory compiling even a small portion of it. It might be possible to prove it for the version of ⌣ explicitly closed under symmetry, which is not a congruence by definition, but would first need to show that ⌣ is a congruence.


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