8.1. Invoking the Simplifier🔗

Lean's simplifier can be invoked in a variety of ways. The most common patterns are captured in a set of tactics. The tactic reference contains a complete list of simplification tactics.

Simplification tactics all contain simp in their name. Aside from that, they are named according to a system of prefixes and suffixes that describe their functionality:

-! suffix

Sets the autoUnfold configuration option to true, causing the simplifier unfold all definitions

-? suffix

Causes the simplifier to keep track of which rules it employed during simplification and suggest a minimal simp set as an edit to the tactic script

-_arith suffix

Enables the use of linear arithmetic simplification rules

d- prefix

Causes the simplifier to simplify only with rewrites that hold definitionally

-_all suffix

Causes the simplifier to repeatedly simplify all assumptions and the conclusion of the goal, taking as many hypotheses into account as possible, until no further simplification is possible

There are two further simplification tactics, simpa and simpa!, which are used to simultaneously simplify a goal and either a proof term or an assumption before discharging the goal. This simultaneous simplification makes proofs more robust to changes in the simp set.

8.1.1. Parameters🔗

The simplification tactics have the following grammar:

syntax
tactic ::= ...
    | The `simp` tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.
- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
  with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.-
- If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated
  with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`.
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
  attribute `[simp]` and all hypotheses.
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
  with the attribute `[simp]`, but removes the ones named `idᵢ`.
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
  the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
  `hᵢ` is introduced, but the old one remains in the local context.
- `simp at *` simplifies all the hypotheses and the target.
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
  other hypotheses.
simp Configuration options for tactics. optConfig only? ([ ((The simp lemma specification `*` means to rewrite with all hypotheses simpStar | An erasure specification `-thm` says to remove `thm` from the simp set simpErase | A simp lemma specification is:
* optional `↑` or `↓` to specify use before or after entering the subterm
* optional `←` to use the lemma backward
* `thm` for the theorem to rewrite with
simpLemma)),* ] )? (Location specifications are used by many tactics that can operate on either the
hypotheses or the goal. It can have one of the forms:
* 'empty' is not actually present in this syntax, but most tactics use
  `(location)?` matchers. It means to target the goal only.
* `at h₁ ... hₙ`: target the hypotheses `h₁`, ..., `hₙ`
* `at h₁ h₂ ⊢`: target the hypotheses `h₁` and `h₂`, and the goal
* `at *`: target all hypotheses and the goal
at A hypothesis location specification consists of 1 or more hypothesis references
and optionally `⊢` denoting the goal.
term*)?

In other words, an invocation of a simplification tactic takes the following modifiers, in order, all of which are optional:

  • A configuration options, which should the fields of Lean.Meta.Simp.Config or Lean.Meta.DSimp.Config, depending on whether the simplifier being invoked is a version of simp or a version of dsimp.

  • The Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or non-dependent hypotheses. It has many variants: - `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`. - `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.- - If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`. - `simp [*]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]` and all hypotheses. - `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas. - `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]`, but removes the ones named `idᵢ`. - `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis `hᵢ` is introduced, but the old one remains in the local context. - `simp at *` simplifies all the hypotheses and the target. - `simp [*] at *` simplifies target and all (propositional) hypotheses using the other hypotheses. only modifier excludes the default simp set, instead beginning with an emptyTechnically, the simp set always includes eq_self and iff_self in order to discharge reflexive cases. simp set.

  • The lemma list adds or removes lemmas from the simp set. There are three ways to specify lemmas in the lemma list:

    • *, which adds all assumptions in the proof state to the simp set

    • - followed by a lemma, which removes the lemma from the simp set

    • A lemma specifier, consisting of the following in sequence:

      • An optional ↓ or ↑, which respectively cause the lemma to be applied before or after entering a subterm (↑ is the default). The simplifier typically simplifies subterms before attempting to simplify parent terms, as simplified arguments often make more rules applicable; ↓ causes the parent term to be simplified with the rule prior to the simplification of subterms.

      • An optional ←, which causes equational lemmas to be used from right to left rather than from left to right.

      • A mandatory lemma, which can be a simp set name, a lemma name, or a term. Terms are treated as if they were named lemmas with fresh names.

  • A location specifier, preceded by Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or non-dependent hypotheses. It has many variants: - `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`. - `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.- - If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`. - `simp [*]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]` and all hypotheses. - `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas. - `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged with the attribute `[simp]`, but removes the ones named `idᵢ`. - `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis `hᵢ` is introduced, but the old one remains in the local context. - `simp at *` simplifies all the hypotheses and the target. - `simp [*] at *` simplifies target and all (propositional) hypotheses using the other hypotheses. at, which consists of a sequence of locations. Locations may be:

    • The name of an assumption, indicating that its type should be simplified

    • An asterisk *, indicating that all assumptions and the conclusion should be simplified

    • A turnstile ⊢, indicating that the conclusion should be simplified

    By default, only the conclusion is simplified.

Location specifiers for simp

In this proof state,

p:Nat → Propx:Nath:p (x + 5 + 2)h':p (3 + x + 9)⊢ p (6 + x + 1)

the tactic p:Nat → Propx:Nath:p (x + 5 + 2)h':p (3 + x + 9)⊢ p (x + 7) simplifies only the goal:

p:Nat → Propx:Nath:p (x + 5 + 2)h':p (3 + x + 9)⊢ p (x + 7)

Invoking p:Nat → Propx:Nath':p (3 + x + 9)h:p (x + 7)⊢ p (6 + x + 1) yields a goal in which the hypothesis h has been simplified:

p:Nat → Propx:Nath':p (3 + x + 9)h:p (x + 7)⊢ p (6 + x + 1)

The conclusion can be additionally simplified by adding ⊢, that is, p:Nat → Propx:Nath':p (3 + x + 9)h:p (x + 7)⊢ p (x + 7):

p:Nat → Propx:Nath':p (3 + x + 9)h:p (x + 7)⊢ p (x + 7)

Using p:Nat → Propx:Nath:p (x + 7)h':p (x + 12)⊢ p (x + 7) simplifies all assumptions together with the conclusion:

p:Nat → Propx:Nath:p (x + 7)h':p (x + 12)⊢ p (x + 7)