simp

Summary

There are many lemmas of the form x = y or P ↔ Q in mathlib which are “tagged” with the @[simp] tag. Note that these kinds of lemmas are ones for which the rw tactic can be used. A lemma tagged with @[simp] is called a “simp lemma”.

When Lean’s simplifier simp is run, it tries to find simp lemmas for which the left hand side of the lemma is in the goal. It then rewrites the lemma and continues. Ultimately what happens is that the goal ends up simplified, and, ideally, solved.

Overview

There are hundreds of lemmas in mathlib of the form x + 0 = x or x * 0 = 0, where one side is manifestly simpler than the other (in the sense that a mathematician would instinctively replace the more complex side by the simpler side if they were trying to prove a theorem). In Lean, such a lemma might be tagged with the @[simp] tag. A tag on a lemma or definition does nothing mathematical; it is just a flag for certain tactics. The convention for @[simp] lemmas in Lean is that the left hand side of such a lemma should be the more complicated side. For example Lean has

@[simp] add_zero (x) : x + 0 = x := ...
@[simp] zero_add (x) : 0 + x = x := ...

(proofs omitted). Because these lemmas are tagged with @[simp], the following proof works:

example (x : ℝ) : 0 + 0 + x + 0 + 0 = (x + (0 + 0)) := by
  simp

If you want to know which lemmas simp used, you can instead use simp?, which gives an output listing the theorems which simp rewrote to make the progress which it made.

Examples

1. If you are doing a proof by induction, then simp will often deal with the base case, because it knows lots of ways to simplify goals with a 0 in. Here is simp being used to prove that \(\sum_{i=0}^{n-1}i=n(n-1)/2\):

import Mathlib.Tactic

open BigOperators Finset -- ∑ notation and access to `Finset.range`

example (n : ℕ) :
   ∑ i in range n, (i : ℝ) = n * (n - 1) / 2 := by
induction' n with d hd
·  -- base case: sum over empty type is 0 * (0 - 1) / 2
   simp
· -- inductive step
   rw [sum_range_succ, hd]
   simp -- tidies up and reduces the goal to
         -- ⊢ ↑d * (↑d - 1) / 2 + ↑d = (↑d + 1) * ↑d / 2
   ring -- a more appropriate tactic to finish the job

Details

Note that simp, like rw, works up to syntactic equality. In other words, if your goal mentions x and there is a simp lemma h : x' = y where x' is definitionally, but not syntactically, equal to x, then simp will not do the rewrite; this is because rw [h] fails.

You can use simp on goals too: simp at h will run the simplifier on h.

There’s quite a lot to say about the simp tactic. More details of how to use it can be found in Theorem Proving In Lean, in the section “using the simplifier” here .

Further notes

The related tactic simp? attempts to figure out exactly which lemmas simp used.