## Integrating Class and Equality Solving

### Current main functions

Equality solver

```tcReduceEqs :: [Inst]             -- locals
-> [Inst]             -- wanteds
-> TcM ([Inst],       -- normalised locals (w/o equalities)
[Inst],       -- normalised wanteds (including equalities)
TcDictBinds,  -- bindings for all simplified dictionaries
Bool)         -- whether any flexibles where instantiated

```

It already processes both equalities and class constraints, but only to simplify the type parameters of class constraints, no attempt to simplify them is made.

Predicate solver

```reduceList :: RedEnv -> [Inst] -> Avails -> TcM Avails
```

The [Inst] currently includes equalities, but they are ignored. The Avails passed in are free of equalities, but the outgoing ones includes equalities (which come from the [Inst]). Moreover, we have the post-processing of Avails by

```extractResults :: Avails
-> [Inst]		-- Wanted
-> TcM (TcDictBinds, 	-- Bindings
[Inst],		-- The insts bound by the bindings
[Inst])		-- Irreducible ones
-- Note [Reducing implication constraints]
```

The irreducible Insts will include all equalities that are passed in as wanteds.

### Integration

Both the equality and class solver have a pre-processing and post-processing phase as well as the main solving phase. (All three phases are structurally more complicated for equalities.)

#### Pre-processing

• Classes: computation of the initial Avails value (called init_state in reduceContext).
• Equalities: normalisation of equalities and class constraints.

I'd suggest to do normalisation first (basically in the same way as it done now) and then build the initial Avails from the normalised constraints.

#### Main solver

We can probably leave the solving of class constraints much as it is now. How much the solving of equalities will have to change depends on how we handle Avails and whether we keep local equalities in Avails, too. (At the moment, the code in TcTyFuns doesn't use the Avails data type.) We need to integrate the loops of TcSimplify.reduceList/TCSimplify.reduce with those of TcTyFuns.propagate.

#### Post-processing

• Classes: Extract bindings, bound insts, and irreducible insts from the final Avails (i.e., work of extractResult).
• Equalities: Finalisation phase.

I'd suggest to extract results first and then finalise on the insts as usual. That way, the code for both probably doesn't have to change much. It's also conceptually simpler, I think.

#### Open questions

1. During solving we maintain the local equalities and local class constraints in rather different structures. In particular, we don't use Avails for equalities. How do we want to integrate this?
2. Integration of loops of TcSimplify.reduceList/TCSimplify.reduce with those of TcTyFuns.propagate.

### Module structure

Currently, the equality solver lives in TcTyFuns and the class solver lives in TcSimplify together with the higher-level logic. This set up is no longer feasible when we integrate both solvers as we don't want to add a recursive module dependency. It seems undesirable to have everything in TcSimplify; so, we should move the class constraint solving out. I think, there are two options:

1. We could have a new module TcSolver that includes all the code currently in TcTyFuns plus all the class constraint solving code from TcSimplify (and remove TcTyFuns.)
2. We could have a new module TcTyPred that gets all the class constraint solver code from TcSimplify. The code that coordinates solving of equalities with class constraints should go into TcTyFuns as it needs to do the preparatory normalising of types (lifting out family synonym applications) and similar things that are due to type families. TcTyFuns will then invoke functions in TcTyPred to perform class solving steps.

To me, the second option is more appealing as I expect it to keep interfaces smaller.

### Functional dependencies

Currently, tcImproveOne generates pairs of types to be unified (on the basis of the FD improvement rules) and does unify them with unifyType (via unifyEqns). In the integrated solver, tcImproveOne should generate equality constraints (with identity coercions) instead. This will get rid of the extra_eqs that we currently have in reduceContext.

### Additional points raised during phone conf, 3 Dec

The plan is to outline on this page, the final design we are aiming for (independent of whether we implement it in one rewrite of the source or more).

#### Treatment of variable instantiation

Instead of instantiating type variables during finalisation, we should already remove equalities of the form co :w alpha ~> t during normalisation or propagation and remember them as substitutions alpha := t in EqConfig (or similar) and instantiate co := id. We also need to ensure that the substitution is propagated to all other constraints.

When we return to reduceContext and solve the implication constraint, we also got to use the substitutions as given equalities when we try to solve the implications.

#### EqConfig

EqConfig needs to become more general and be a general solver configuration. Likewise RewriteInst needs to include a variant for class constraints, which we need to handle during propagate.