The `simp` tactic uses a discrimination tree to select candidate
theorems that will be used to rewrite an expression. This indexing data
structure minimizes the number of theorems that need to be tried and
improves performance. However, indexing modulo reducibility is
challenging, and a theorem that could be applied, when taking reduction
into account, may be missed. For example, suppose we have a `simp`
theorem `foo : forall x y, f x (x, y).2 = y`, and we are trying to
simplify the expression `f a b <= b`. `foo` will not be tried by `simp`
because the second argument of `f a b` is not a projection of a pair.
However, `f a b` is definitionally equal to `f a (a, b).2` since we can
reduce `(a, b).2`.
In Lean 3, we had a much simpler indexing data structure where only the
head symbol was taken into account. For the theorem `foo`, the head
symbol is `f`. Thus, the theorem would be considered by `simp`.
This commit adds the option `Simp.Config.index`. When `simp (config := {
index := false })`, only the head symbol is considered when retrieving
theorems, as in Lean 3. Moreover, if `set_option diagnostics true`,
`simp` will check whether every applied theorem would also have been
applied if `index := true`, and report them. This feature can help users
diagnose tricky issues in code that has been ported from libraries
developed using Lean 3 and then ported to Lean 4. In the following
example, it will report that `foo` is a problematic theorem.
```lean
opaque f : Nat → Nat → Nat
@[simp] theorem foo : f x (x, y).2 = y := by sorry
example : f a b ≤ b := by
set_option diagnostics true in
simp (config := { index := false })
```
In the example above, the following diagnostic message is produced.
```lean
[simp] theorems with bad keys
foo, key: [f, *, Prod.1, Prod.mk, Nat, Nat, *, *]
```
With the information above, users can annotate theorems such as `foo`
using `no_index` for problematic subterms.
Example:
```lean
opaque f : Nat → Nat → Nat
@[simp] theorem foo : f x (no_index (x, y).2) = y := by sorry
example : f a b ≤ b := by
simp -- `foo` is still applied
```
cc @semorrison
cc @PatrickMassot
