## Introduce the `apply_congr`

conv mode tactic. #

`apply_congr`

will apply congruence lemmas inside `conv`

mode.
It is particularly useful when the automatically generated congruence lemmas
are not of the optimal shape. An example, described in the doc-string is
rewriting inside the operand of a `Finset.sum`

.

Apply a congruence lemma inside `conv`

mode.

When called without an argument `apply_congr`

will try applying all lemmas marked with `@[congr]`

.
Otherwise `apply_congr e`

will apply the lemma `e`

.

Recall that a goal that appears as `∣ X`

in `conv`

mode
represents a goal of `⊢ X = ?m`

,
i.e. an equation with a metavariable for the right hand side.

To successfully use `apply_congr e`

, `e`

will need to be an equation
(possibly after function arguments),
which can be unified with a goal of the form `X = ?m`

.
The right hand side of `e`

will then determine the metavariable,
and `conv`

will subsequently replace `X`

with that right hand side.

As usual, `apply_congr`

can create new goals;
any of these which are *not* equations with a metavariable on the right hand side
will be hard to deal with in `conv`

mode.
Thus `apply_congr`

automatically calls `intros`

on any new goals,
and fails if they are not then equations.

In particular it is useful for rewriting inside the operand of a `Finset.sum`

,
as it provides an extra hypothesis asserting we are inside the domain.

For example:

```
example (f g : ℤ → ℤ) (S : Finset ℤ) (h : ∀ m ∈ S, f m = g m) :
Finset.sum S f = Finset.sum S g := by
conv_lhs =>
-- If we just call `congr` here, in the second goal we're helpless,
-- because we are only given the opportunity to rewrite `f`.
-- However `apply_congr` uses the appropriate `@[congr]` lemma,
-- so we get to rewrite `f x`, in the presence of the crucial `H : x ∈ S` hypothesis.
apply_congr
· skip
· simp [*]
```

In the above example, when the `apply_congr`

tactic is called it gives the hypothesis `H : x ∈ S`

which is then used to rewrite the `f x`

to `g x`

.

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