The idea is to match the precedence used in regular programming
languages, where `x = y || x = z` is parsed as `(x = y) || (x = z)`.
This commit also adds `!x` as notation for `bnot x`
With the current elaboration scheme, out_params and coercions do not mix well,
as evidenced by the following example by @digama:
```
variables {α : Type*} [group α]
def gpow : α → ℤ → α := sorry
instance group.has_pow : has_pow α ℤ := ⟨gpow⟩
example (a : α) : a ^ 0 = 1 := sorry -- failed to synth ⊢ has_pow α ℕ
example (a : α) : a ^ (0:ℕ) = 1 := sorry -- ok, coerces
example (a : α) : a ^ (0:ℤ) = 1 := sorry -- ok
```
The issue is that
* we first try to solve `has_pow ?α ?β`, which is postponed
* then infer `?α = nat` from `a`
* then at some point call `elaborator::synthesize()` and default `β` to `nat`
* then try to solve `has_pow nat nat`, which fails at `int =?= nat`
This command is not just a cosmetic feature.
We need it to defined `id_rhs` before the tactic framework is defined.
We want `id_rhs` to be used in all definitions generated by the equation
compiler. Right now, it is only used in definitions defined after the
tactic framework.
@kha: I decided to implement this change before I start the
type_context modifications. The change did not affect the corelib and
test suite much. The only annoying problem is that `out` cannot be
used to name locals anymore.
This way people can search for "constant quot" and find it in the lean source. Plus the init_quotient command only occurs once, so this way people know what it means.
See Section "Other goodies" at
https://github.com/leanprover/lean/wiki/Refactoring-structures
This commit also improves the support for projections in the
unifier/matcher.
Now, we consider the extra case-split for projections.
Given a projection `proj`, and the constraint `proj s =?= proj t`, we need to try first `s =?= t` and if it fails, then try to reduce.
This is needed in the standard library because we now have constraints such as:
```
@has_le.le ?A ?s ?a ?b =?= @has_le.le nat nat.has_add x y
```
If we reduce the right hand side, we get the unsolvable constraint
```
@has_le.le ?A ?s ?a ?b =?= nat.le x y
```
Before this change, the constraint was `@le ?A ?s ?a ?b =?= @le nat nat.has_add x y`, and we already perform a case-split in this case.
Moreover, projections were eagerly reduced whenever possible.
The extra case-split generates a performance problem in several tests. For example `fib 8 = 34` was timing out.
I worked around this issue by performing the case-split only when the constraint contains meta-variables.
There are also minor issues. Example. `<` is notation for `has_lt.lt`, but `>` is for `gt`.
