Now, the elaborator only uses the quasi-pattern unifier approximation
for inferring the implicit motive in recursor-like applications.
This change was motivated by counterintuitive behavior associated with
this approximation. For example, before this commit
```
variables {δ σ : Type}
def ex1 : state_t δ (state_t σ id) σ :=
monad_lift (get : state_t σ id σ) -- doesn't work
def ex2 : state_t δ (state_t σ id) σ :=
do s ← monad_lift (get : state_t σ id σ), -- works
return s
```
The first one doesn't work because when we elaborate
`@monad_lift ?m ?n ?c ?α (get : state_t σ id σ) : ?n ?α`
with expected type `state_t δ (state_t σ id) σ`
It first produces the following unification problem by processing
matching the inferred type with the expected one.
```
?n ?α =?= state_t δ (state_t σ id) σ
==> (approximate using first-order unification)
?n := state_t δ (state_t σ id)
?α := σ
```
Then we try to solve
```
?m ?α =?= state_t σ id σ
==> instantiate metavars
?m σ =?= state_t σ id σ
==> (approximate since it is a quasi-pattern unification constraint)
?m := λ σ, state_t σ id σ
```
Remark: the constraint is not a Milner pattern because `σ` is in
the local context of `?m`. By assuming it is a Milner pattern,
we are ignoring the other possible solutions:
```
?m := λ σ', state_t σ id σ
?m := λ σ', state_t σ' id σ
?m := λ σ', state_t σ id σ'
```
We need the quasi-pattern approximation for elaborating recursors.
So, this commit enable this kind of approximation only when
elaborating recursors and executing induction-like tactics.
If we had used first-order unification, then we would have produced
the right answer: `?m := state_t σ id`
Haskell would solve this example since it always uses
first-order unification during elaboration.
The second one works because when we elaborate
`monad_lift (get : state_t σ id σ)`, the expected type is `state_t δ (state_t σ id) ?α`.
So, `?m ?α =?= state_t σ id σ` will not considered to be a quasi-pattern
since `?α` is not yet assigned to a local constant.
We are not fully confident this commit produces a better user
experience. We know that
- Full higher-order unification (used in Lean2) produces a combinatoric
explosion, and generates a lot of non-termination in complex type class
hierarchies (monad library, has_coe, etc). The problem is that
higher-order unification manages to create new solutions that we
cannot find using first-order unification.
- Lean3 is more reliable than Lean2 for elaborating monadic code because
it does not use higher-order unification.
- For elaborating recursor-like applications, we need at least the
quasi-patterns. We need it when trying to infer the implicit
motive. First-order unification works poorly in this case. Note that
the lack of higher-order unification in Lean3 forces us to provide the
motive explicitly for terms that Lean2 can elaborate.
- We need quasi-patterns for solving unification constraints in the
induction-like tactics. Similar to the previous item. We use it to infer
the motive. (edited) I will try to disable the quasi-pattern
approximation when elaborating regular applications. At least, we will
behave like Haskell for this kind of application.
Before this commit, the unifier would try to solve the unification consraint
?m =?= fun x_1 ... x_n, ?m x_1 ... x_n
by assigning
?m := fun x_1 ... x_n, ?m x_1 ... x_n
which fails the occurs check.
This commit skips the assignment by using eta-reduction.
@nunoplopes @aqjune
I had to add a new primitive to allow you to execute a tactic from the
`main` function. The `main` function is in the `io` monad. The new
primitive has type:
```
meta constant io.run_tactic {α : Type} (a : tactic α) : io α
```
I also added a new test that shows how to use it.
The test displays all declarations that have the `nat` prefix.
cc @kha
Reason: `rsimp` is based on the smt framework. The smt framework
has to be reimplemented. Moreover, the smt framework is currently
not using the new cache infrastructure and we pay a substantial
performance penalty.
