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# Entailment of Type Equalities

Our aim is to derive the entailment judgement

g1, .., gn |- w1, .., wm

i.e., whether we can derive the wanted equalities `w1` to `wm` from the given equalities `g1`, .., `gn` under a given set of toplevel equality schemas (i.e., equalities involving universally quantified variables). We permit unification variables in the wanted equalities, and a derivation may include the instantiation of these variables; i.e., may compute a unifier. However, that unifer must be most general.

The derivation algorithm is complicated by the pragmatic requirement that, even if there is no derivation for the judgement, we would like to compute a unifier. This unifier should be as specific as possible under some not yet specified (strongest) weakening of the judgement so that it is derivable. (Whatever that means...)

The following is based on ideas for the new, post-ICFP'08 solving algorithm described in CVS `papers/type-synonym/new-single.tex`. A revised version of `new-single.tex` that integrates the core ideas from this wiki page is in `papers/type-synonym/normalised_equations_algorithm.tex`. Most of the code is in the module `TcTyFuns`.

## Terminology

*Wanted equality*- An equality constraint that we need to derive during type checking. Failure to derive it leads to rejection of the checked program.
*Local equality*,*given equality*- An equality constraint that -in a certain scope- may be used to derive wanted equalities.
*Flexible type variable*,*unification variable*,*HM variable*-
Type variables that may be
**globally**instantiated by unification. We use Greek letters`alpha, beta,`... as names for these variables. *Rigid type variable*,*skolem type variable*-
Type variable that cannot be globally instantiated, but it may be
**locally**refined by a local equality constraint. We use Roman letters`a, b,`... as names for these variables.

In positions where we can have both flexible and rigid variables, we use `x, y, z`.

## Overall algorithm

The overall structure is as in `new-single.tex`, namely

- normalise all constraints (both locals and wanteds),
- solve the wanteds, and
- finalise.

However, the three phases differ in important ways. In particular, normalisation includes decompositions & the occurs check, and we don't instantiate any flexible type variables before we finalise (i.e., solving is purely local).

## Normal equalities

Central to the algorithm are **normal equalities**, which can be regarded as a set of rewrite rules. Normal equalities are carefully oriented and contain synonym families only as the head symbols of left-hand sides. They assume one of the following two major forms:

**Family equality:**`co :: F t1..tn ~ t`or**Variable equality:**`co :: x ~ t`, where we again distinguish two forms:**Variable-term equality:**`co :: x ~ t`, where`t`is*not*a variable, or**Variable-variable equality:**`co :: x ~ y`, where`x > y`.

where

- the types
`t`,`t1`, ...,`tn`may not contain any occurrences of synonym families, - the left-hand side of an equality may not occur in the right-hand side, and
- the relation
`x > y`is a total order on type variables, where`alpha > a`whenever`alpha`is a flexible and`a`a rigid type variable (otherwise, the total order may be aribitrary).

The second bullet of the where clause is trivially true for equalities of Form (1); it also implies that the left- and right-hand sides are different.

Furthermore, we call a variable equality whose left-hand side is a flexible type variable (aka unification variable) a **flexible variable equality**, and correspondingly, a variable equality whose left-hand side is a rigid type variable (aka skolem variable) a **rigid variable equality.**

### Observations

The following is interesting to note:

- Normal equalities are similar to equalities meeting the Orientation Invariant and Flattening Invariant of new-single, but they are not the same.
- Normal equalities are
**never**self-recursive. They can be mutually recursive. A mutually recursive group will exclusively contain variable equalities.

### Coercions

Coercions `co` are either wanteds (represented by a flexible type variable) or givens *aka* locals (represented by a type term of kind CO). In GHC, they are represented by `TcRnTypes.EqInstCo`, which is defined as

type EqInstCo = Either TcTyVar -- case for wanteds (variable to be filled with a witness) Coercion -- case for locals

Moreover, `TcTyFuns.RewriteInst` represents normal equalities, emphasising their role as rewrite rules.

**SLPJ**: I propose that we use a proper data type, not `Either` for this.

## Normalisation

The following function `norm` turns an arbitrary equality into a set of normal equalities. As in `new-single`, the evidence equations are differently interpreted depending on whether we handle a wanted or local equality.

data EqInst -- arbitrary equalities data FlatEqInst -- synonym families may only occur outermost on the lhs data RewriteInst -- normal equality norm :: EqInst -> [RewriteInst] norm [[co :: F s1..sn ~ t]] = [[co :: F s1'..sn' ~ t']] : eqs1++..++eqsn++eqt where (s1', eqs1) = flatten s1 .. (sn', eqsn) = flatten sn (t', eqt) = flatten t norm [[co :: t ~ F s1..sn]] = norm [[co' :: F s1..sn ~ t]] with co = sym co' norm [[co :: s ~ t]] = check [[co :: s' ~ t']] ++ eqs ++ eqt where (s', eqs) = flatten s (t', eqt) = flatten t check :: FlatEqInst -> [RewriteInst] -- Does OccursCheck + Decomp + Triv + Swap (of new-single) check [[co :: t ~ t]] = [] with co = id check [[co :: x ~ y]] | x > y = [[co :: x ~ y]] | otherwise = [[co' :: y ~ x]] with co = sym co' check [[co :: x ~ t]] | x `occursIn` t = fail | otherwise = [[co :: x ~ t]] check [[co :: t ~ x]] | x `occursIn` t = fail | otherwise = [[co' :: x ~ t]] with co = sym co' check [[co :: s1 s2 ~ t1 t2]] = check [[col :: s1 ~ t1]] ++ check [[cor :: s2 ~ t2]] with co = col cor check [[co :: T ~ S]] = fail flatten :: Type -> (Type, [RewriteInst]) -- Result type has no synonym families whatsoever flatten [[F t1..tn]] = (alpha, [[id :: F t1'..tn' ~ alpha]] : eqt1++..++eqtn) where (t1', eqt1) = flatten t1 .. (tn', eqtn) = flatten tn FRESH alpha, such that alpha > x for all x already used RECORD alpha := F t1'..tn' flatten [[t1 t2]] = (t1' t2', eqs++eqt) where (t1', eqs) = flatten t1 (t2', eqt) = flatten t2 flatten t = (t, [])

The substitutions `RECORD`ed during `flatten` need to be (unconditionally) applied during finalisation (i.e., the 3rd phase).

Notes:

- Perform Rule Triv as part of normalisation.
- Whenever an equality of Form (2) or (3) would be recursive, the program can be rejected on the basis of a failed occurs check. (Immediate rejection is always justified, as right-hand sides do not contain synonym familles; hence, any recursive occurrences of a left-hand side imply that the equality is unsatisfiable.)
- We flatten locals and wanteds in the same manner, using fresh flexible type variables. (We have flexibles in locals anyway and don't use (Unify) during normalisation - this is different to
`new-single`.)

**TODO** We may want to drop the `alpha > x for all x already used` side condition on fresh `alpha`s again. Instead, we should probably consider all *local* flexible variable equalities during finalisation, too. At least, those involving flexible variables introduced by flattening.

## Propagation (aka Solving)

Propagation is based on four rules that transform family and variable equalities. The first rule transforms a family equality using a toplevel equality schema. The other three use one equality to transform another. In the presentation, the transforming equality is always first and the transformed is second, separate by an ampersand (`&`).

Below the rules are two scheme for applying the rules. The first one naively iterates until the system doesn't change anymore. The second scheme optimises the iteration somewhat.

### Rules

**Top**-
co :: F t1..tn ~ t =(Top)=> co' :: [s1/x1, .., sm/xm]s ~ t with co = g s1..sm |> co'

where`g :: forall x1..xm. F u1..um ~ s`and`[s1/x1, .., sm/xm]u1 == t1`.

NB: Afterwards, we need to normalise. Then, any of the four propagation rules may apply.

**SubstFam**-
co1 :: F t1..tn ~ t & co2 :: F t1..tn ~ s =(SubstFam)=> co1 :: F t1..tn ~ t & co2' :: t ~ s with co2 = co1 |> co2'

where`co1`is local, or both`co1`and`co2`are wanted and at least one of the equalities contains a flexible variable.

NB: Afterwards, we need to normalise

co2'. Then, the SubstVarVar or SubstVarFam rules may apply to the results of normalisation. Moreover,co1may have a SubstFam or a SubstVarFam match with rules other than the results of normalisingco2'.

**SubstVarVar**-
co1 :: x ~ t & co2 :: x ~ s =(SubstVarVar)=> co1 :: x ~ t & co2' :: t ~ s with co2 = co1 |> co2'

where`co1`is local, or both`co1`and`co2`are wanted and at least one of the equalities contains a flexible variable.

NB: Afterwards, we need to normalise

co2'. Then, the SubstVarVar or SubstVarFam rules may apply to the results of normalisation, but not withco1, assandtcannot containx-- cf. the definition of normal equalities. However,co1may have another SubstVarVar or SubstVarFam match with rules other than the results of normalisingco2'.

**SubstVarFam**-
co1 :: x ~ t & co2 :: F s1..sn ~ s =(SubstVarFam)=> co1 :: x ~ t & co2' :: [t/x](F s1..sn) ~ s with co2 = [co1/x](F s1..sn) |> co2'

where`x`occurs in`F s1..sn`. (`co1`may be local or wanted.)

NB: No normalisation required. Afterwards, SubstVarVar or SubstVarFam may apply to

co1and all rules, except SubstVarVar, may apply toco2'. However, SubstVarFam cannot again apply toco1andco2', astcannot containx-- cf. the definition of normal equalities..

### Rule application: specification

Propagation proceeds by applying any of the four rules until the system does not change anymore. After application of a rule, the equalities that were modified need to be normalised again:

Propagate = fix (Top | SubstFam | SubstVarVar | SubstVarFam)

### Rule application: algorithm

The following function gets a list with all local and wanted equalities. It returns a list of residual equalities that permit no further rule application.

-- all four rule functions return Nothing if rule not applicable applyTop :: RewriteInst -> Maybe EqInst applySubstFam :: RewriteInst -> RewriteInst -> Maybe EqInst -- return only... applySubstVarVar :: RewriteInst -> RewriteInst -> Maybe EqInst -- ...the modified... applySubstVarFam :: RewriteInst -> RewriteInst -> Maybe EqInst -- ...equality propagate :: [RewriteInst] -> [RewriteInst] propagate eqs = snd (prop eqs []) where prop :: [RewriteInst] -> [RewriteInst] -> [RewriteInst] prop [] res = res prop (eq:eqs) res = apply eq eqs res apply eq@[[co :: F t1..tn ~ t]] eqs res | Just eq' <- applyTop eq = prop (norm eq' ++ eqs) res | otherwise = mapRule (applySubstFam eq) -- TODO!!!

### Observations

- Only SubstVarFam when replacing a variable in a family equality can lead to recursion with (Top).
- A significant difference to
`new-single`is that solving is a purely local operation. We never instantiate any flexible variables. - We cannot discard any variable equalities after substitution since we do not substitute into right-hand sides anymore. Moreover, in the concrete implementation, variables may also be re-introduced due to simplification of type classes (which currently doesn't happen in the same iteration).

## Finalisation

If only flexible type equalities remain as wanted equalities, the locals entail the wanteds. We can now instantiate type variables in flexible type equalities where possible to propagate constraints into the environment. In GHC, we may wrap any remaining equalities (of any form) into an implication constraint to be propagated outwards (where it may be solved under an extended set of local equalities.)

Notes:

- (Unify) is an asymmetric rule, and hence, only fires for equalities of the form
`x ~ c`, where`c`is free of synonym families. Moreover, it only applies to wanted equalities. (Rationale: Local equality constraints don't justify global instantiation of flexible type variables - just as in new-single.)

**TODO** Still need to adapt example to new rule set!

**TODO** Now that we delay instantiation until after solving, do we still need to prioritise flexible variables equalities over rigid ones? (Probably not.) In any case, we need to handle variable-variable equalities (at least of flexibles) specially during finalisation. Even if they locals, they need to be taken into account during finalisation. (Otherwise, we may miss some instantiations of flexibles that occur in the environment.)

## Examples

### Substituting wanted family equalities with SubstFam is crucial if the right-hand side contains a flexible type variable

Top: F Int ~ [Int] |- F delta ~ [delta], F delta ~ [Int] (SubstFam) |- F delta ~ [delta], norm [[ [delta] ~ [Int] ]] ==> |- F delta ~ [delta], delta ~ Int (SubstVar) |- norm [[ F Int ~ [Int] ]], delta ~ Int ==> |- F Int ~ [Int], delta ~ Int (Top) |- norm [[ [Int] ~ [Int] ]], delta ~ Int ==> |- delta ~ Int QED

### Interaction between local and wanted family equalities

Example 4 of Page 9 of the ICFP'09 paper.

F [Int] ~ F (G Int) |- G Int ~ [Int], H (F [Int]) ~ Bool =(norm)=> F [Int] ~ a, F b ~ a, G Int ~ b |- G Int ~ [Int], H x ~ Bool, F [Int] ~ x (SubstFam w/ F [Int]) F [Int] ~ a, F b ~ a, G Int ~ b |- G Int ~ [Int], H x ~ Bool, x ~ a (SubstFam w/ G Int) F [Int] ~ a, F b ~ a, G Int ~ b |- b ~ [Int], H x ~ Bool, x ~ a (SubstVar w/ x) F [Int] ~ a, F b ~ a, G Int ~ b |- b ~ [Int], H a ~ Bool, x ~ a

**TODO** If we use flexible variables for the flattening of the wanteds, too, the equality corresponding to `x ~ a` above will be oriented the other way around. That can be a problem because of the asymmetry of the SubstVar and SubstFun rules (i.e., wanted equalities are not substituted into locals).

## Termination

### SkolemOccurs

The Note [skolemOccurs loop] in the old code explains that equalities of the form `x ~ t` (where `x` is a flexible type variable) may not be used as rewrite rules, but only be solved by applying Rule Unify. As Unify carefully avoids cycles, this prevents the use of equalities introduced by the Rule SkolemOccurs as rewrite rules. For this to work, SkolemOccurs also had to apply to equalities of the form `a ~ t[[a]]`. This was a somewhat intricate set up that we seek to simplify here. Whether equalities of the form `x ~ t` are used as rewrite rules or solved by Unify doesn't matter anymore. Instead, we disallow recursive equalities after normalisation completely (both locals and wanteds). This is possible as right-hand sides are free of synonym families.

To look at this in more detail, let's consider the following notorious example:

E_t: forall x. F [x] ~ [F x] [F v] ~ v ||- [F v] ~ v

**New-single**: The following derivation shows how the algorithm in new-single fails to terminate for this example.

[F v] ~ v ||- [F v] ~ v ==> normalise v ~ [a], F v ~ a ||- v ~ [x], F v ~ x a := F v ==> (Local) with v F [a] ~ a ||- [a] ~ [x], F [a] ~ x ==> normalise F [a] ~ a ||- x ~ a, F[a] ~ x ==> 2x (Top) & Unify [F a] ~ a ||- [F a] ~ a ..and so on..

**New-single using flexible tyvars to flatten locals, but w/o Rule (Local) for flexible type variables**: With (SkolemOccurs) it is crucial to avoid using Rule (Local) with flexible type variables. We can achieve a similar effect with new-single if we (a) use flexible type variables to flatten local equalities and (b) at the same time do not use Rule (Local) for variable equalities with flexible type variables. NB: Point (b) was necessary for the ICFP'08 algorithm, too.

[F v] ~ v ||- [F v] ~ v ==> normalise v ~ [x2], F v ~ x2 ||- v ~ [x1], F v ~ x1 ** x2 := F v ==> (Local) with v F [x2] ~ x2 ||- [x2] ~ [x1], F [x2] ~ x1 ** x2 := F v ==> normalise F [x2] ~ x2 ||- x2 ~ x1, F [x2] ~ x1 ** x2 := F v ==> 2x (Top) & Unify [F x1] ~ x1 ||- [F x1] ~ x1 ** x1 := F v ==> normalise x1 ~ [y2], F x1 ~ y2 ||- x1 ~ [y1], F x1 ~ y1 ** x1 := F v, y2 := F x1 ..we stop here if (Local) doesn't apply to flexible tyvars

A serious disadvantage of this approach is that we **do** want to use Rule (Local) with flexible type variables as soon as we have rank-n signatures. In fact, the lack of doing so is responsible for a few failing tests in the testsuite in the GHC implementation of (SkolemOccurs).

**De-prioritise Rule (Local)**: Instead of outright forbidding the use of Rule (Local) with flexible type variables, we can simply require that Local is only used if no other rule is applicable. (That has the same effect on satisfiable queries, and in particular, the present example.)

[F v] ~ v ||- [F v] ~ v ==> normalise v ~ [a], F v ~ a ||- v ~ [x], F v ~ x a := F v ==> (IdenticalLHS) with v & F v v ~ [a], F v ~ a ||- [a] ~ [x], x ~ a ==> normalise v ~ [a], F v ~ a ||- x ~ a, x ~ a ==> (Unify) v ~ [a], F v ~ a ||- a ~ a ==> normalise v ~ [a], F v ~ a ||- QED

In fact, it is sufficient to de-prioritise Rule (Local) for variable equalities (if it is used for other equalities at all):

[F v] ~ v ||- [F v] ~ v ==> normalise v ~ [a], F v ~ a ||- v ~ [x], F v ~ x a := F v ==> (Local) with F v v ~ [a], F v ~ a ||- v ~ [x], x ~ a ==> (Unify) v ~ [a], F v ~ a ||- v ~ [a] ==> (Local) with v v ~ [a], F [a] ~ a ||- [a] ~ [a] ==> normalise v ~ [a], F [a] ~ a ||- QED

**One problems remains**: The algorithm still fails to terminate for unsatisfiable queries.

[F v] ~ v ||- [G v] ~ v ==> normalise v ~ [a], F v ~ a ||- v ~ [x], G v ~ x a := F v ==> (Local) with v F [a] ~ a ||- [a] ~ [x], G [a] ~ x ==> normalise F [a] ~ a ||- x ~ a, G [a] ~ x ==> (Unify) F [a] ~ a ||- G [a] ~ a ==> (Top) [F a] ~ a ||- G [a] ~ a ==> normalise a ~ [b], F a ~ b ||- G [a] ~ a b := F a ..and so on..

My guess is that the algorithm terminates for all satisfiable queries. If that is correct, the entailment problem that the algorithm solves would be semi-decidable.