Most efficient hash functions use uint32/uint64 and produce values
that do not fit in out small nat representation. Thus, GMP big numbers
would have to be created.
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`.
