|Version 5 (modified by diatchki, 5 months ago)|
Connection with GHC's Constraint Solver
The solver for the type nats is implemented as an extra stage in GHC's constrraint solver (see TcInteract.thePipeline).
The solver produces evidence (i.e., proofs) when computing new "given" constraints, or when solving existing "wanted" constraints. The evidence is constructed by applications of a set of pre-defined rules. The rules are values of type TypeRep.CoAxiomRule. Conceptually, rules have the form:
name :: forall tyvars. assumptions => conclusion
The rules have the usual logical meaning: the variables are universally quantified, and the assumptions imply the concluson. As a concrete example, consider the rule for left-cancellation of addtion:
AddCanceL :: forall a b c d. (a + b ~ d, a + c ~ d) => b ~ c
The type CoAxiomRule also supports infinte literal-indexed families of simple axioms using constructor CoAxiomTyLit. These have the form:
name(l_1 .. l_n) :: conclusion
In this case conclusion is an equation that contains no type variables but may depend on the literals in the name of the family. For example, the basic definitional axiom for addition, TcTypeNatsRules.axAddDef, uses this mechanism:
AddDef(2,3) :: 2 + 3 ~ 5
At present, the assumptions and conclusion of all rules are equations between types but this restriction is not important and could be lifted in the future.
The rules used by the solver are in module TcTypeNatsRules.
The entry point to the solver is TcTypeNats.typeNatStage.
We start by examining the constraint to see if it is obviously unsolvable (using function impossible), and if so we stash it in the constraint-solver's state and stop. Note that there is no assumption that impossible is complete, but it is important that it is sound, so if impossible returns True, then the constraint is definitely unsolvable, but if impossible returns False, then we don't know if the constraint is solvable or not.
The rest of the stage proceeds depending on the type of constraint, as follows.
Given constraints correspond to adding new assumptions that may be used by the solver. We start by checking if the new constraint is trivial (using function solve). A constraint is considered to be trivial if it matches an already existing constraint or a rule that is known to the solver. Such given constraints are ignored because they do not contribute new information. If the new given is non-trivial, then it will be recorded to the inert set as a new fact, and we proceed to "interact" it with existing givens, in the hope of computing additional useful facts (function computeNewGivenWork).
IMPORTANT: We assume that "given" constraints are processed before "wanted" ones. A new given constraint may be used to solve any existing wanted, so every time we added a new given to the inert set we should move all potentially solvable "wanted" constraint from the inert set back to the work queue. We DON'T do this, because it is quite inefficient: there is no obvious way to compute which "wanted"s might be affected, so we have to restart all of them!
The heart of the interaction is the function interactCt, which performs one step of "forward" reasoning. The idea is to compute new constraints whose proofs are made by an application of a rule to the new given, and some existing givens. These new constraints are added as new work, to be processed further on the next iteration of GHC's constraint solver.
Aside: when we compute the new facts, we check to see if any are obvious contradictions. This is not strictly necessary because they would be detected on the next iteration of the solver. However, by doing the check early we get slightly better error messages because we can report the original constraint as being unsolvable (it leads to a contradiction), which tends to be easier to relate to the original program. Of course, this is not completely fool-proof---it is still possible that a contradiction is detected at a later iteration. An alternative idea---not yet implemented---would be to examine the proof of a contradiction and extract the original constraints that lead to it in the first place.
Derived constraints are facts that are implied by the constraints in the inert set. They do not have complete proofs because they may depend on proofs of as yet unsolved wanted constraints. GHC does not associate any proof terms with derived constraints (to keep things simple?). In the constraint solver, they are mostly used as "hints". For example, consider the wanted constraint 5 + 3 ~ x, where x is a free unification variable. These are the steps we'll take to solve the constraint:
Rules: Add_def(5,3) : 5 + 3 ~ 8 Add_fun : forall a b c1 c2. (a + b ~ c1, a + b ~ c2) => c1 ~ c2 1. Add to inert set: [W] C: 5 + 3 ~ x 2. Generate new derived: [D] Add_fun(C,Add_def) : x ~ 8 (proof discarded) 3. GHC uses this hint to improve and reconsider the wanted: [W] C: 5 + 3 ~ 8 4. Solved: [W] C = Add_def(5,3)
The type-nat solver processes derived constraints in a similar fashion to given constraints: it checks to see if they are trivially known and, if not, then it tries to generate some additional derived constraints. The main difference is that derived constraints can be interacted with all existing constraints to produce new facts, while given constraints only interact with other givens.