Composing Monads

Research Report on Composing Monads for an University project

Report - February 2019

The report is written in Latex and a final version of it has been added to the repo as main.pdf. This work is based on Composing monads* by Mark P. Jones and Luc Duponcheel.

My works is just a study of the main one, without many differences. The main one is the proof that a natural join doesn't exists. I found out that the proof presented in the original paper is wrong, so I presented my own version.

Formalization - February 2020

One year later, I decided to formalize in Lean this work, under the supervision of Dr. Jasmin Blanchette.

The work can easily be found in the src directory.

dir structure

The work is divided into two main namespace: category and composed.

The first one includes the basic structure of functor, premonad and monad. I've decided to built these structures on my own. Anyway, my environment is totally equivalent to the one inside the Lean standard library (the proof of this statement can be found in src/category/conjunction.lean).

The last one groups all the classes and functions required to compose two generic monads (with all the necessary auxiliary functions). With this structure you can safely compose monads without losing any monad property. They are automatically inferred from the two given monads.

src
|- data/maybe
   |- basic.lean
   |- instances.lean
|- category
   |- functor.lean
   |- premonad.lean
   |- monad.lean
   |- lawful.lean
   |- conjunction.lean
   |- natural_join
      |- inv.lean
      |- natural_join.lean
      |- type.lean
|- composed
   |- lawful
      |- prod
      |- dorp
      |- swap
   |- basic
   |- prod
   |- dorp
   |- swap
   |- instances

data/maybe is just an instance example of my own monadic environment.
category/lawful exposes the lawful classes for functor, premonad and monad. They slightly differ from the lawful described in the standard library, but as shown in category/conjunction they are equivalent.
category/conjunction, this module defines the equivalence between the standard and the custom monadic structure.
category/natural_join is the proof that a function with the right sign (MNMNX → MNX) cannot be found only with the given monads M and N. inv is the invariant that every function has to satisfy, and type is the system that abstract over the real Lean typing system to permit us to use pattern matching over the monad type.
composed/basic defines the structure for the composition of functor and premonad only. composed/prod, composed/dorp and composed/swap define the structure for the monad composition instead.
composed/instances prove that the composition of two functors or premonads respect their lawful classes.
composed/lawful/* prove that the composition of two monads (plus an auxiliary function) respects it monadic lawful class.

Note that some marginal proofs aren't complete, of course the main one is totally done (natural join doesn't exists).

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