Simps attribute

This file defines the @[simps] attribute, to automatically generate simp lemmas reducing a definition when projections are applied to it.

Implementation Notes

There are three attributes being defined here

• @[simps] is the attribute for objects of a structure or instances of a class. It will automatically generate simplification lemmas for each projection of the object/instance that contains data. See the doc strings for Lean.Parser.Attr.simps and Simps.Config for more details and configuration options.
• structureExt (just an environment extension, not actually an attribute) is automatically added to structures that have been used in @[simps] at least once. This attribute contains the data of the projections used for this structure by all following invocations of @[simps].
• @[notation_class] should be added to all classes that define notation, like Mul and Zero. This specifies that the projections that @[simps] used are the projections from these notation classes instead of the projections of the superclasses. Example: if Mul is tagged with @[notation_class] then the projection used for Semigroup will be fun α hα ↦ @Mul.mul α (@Semigroup.toMul α hα) instead of @Semigroup.mul. [this is not correctly implemented in Lean 4 yet]

Possible Future Improvements

• If multiple declarations are generated from a simps without explicit projection names, then only the first one is shown when mousing over simps.

Changes w.r.t. Lean 3

There are some small changes in the attribute. None of them should have great effects

• The attribute will now raise an error if it tries to generate a lemma when there already exists a lemma with that name (in Lean 3 it would generate a different unique name)
• transparency.none has been replaced by TransparencyMode.reducible
• The attr configuration option has been split into isSimp and attrs (for extra attributes)
• Because Lean 4 uses bundled structures, this means that simps applied to anything that implements a notation class will almost certainly require a user-provided custom simps projection.

Tags

structures, projections, simp, simplifier, generates declarations

def updateName (nm : Lean.Name) (s : String) (isPrefix : Bool) : Lean.Name

updateName nm s isPrefix adds s to the last component of nm, either as prefix or as suffix (specified by isPrefix), separated by _. Used by simps_add_projections.

Equations

• updateName nm s isPrefix = Lean.Name.updateLast (fun (s' : String) => if isPrefix = true then s ++ "_" ++ s' else s' ++ "_" ++ s) nm

def Lean.Meta.mkSimpContextResult (cfg : optParam Lean.Meta.Simp.Config { maxSteps := Lean.Meta.Simp.defaultMaxSteps, maxDischargeDepth := 2, contextual := false, memoize := true, singlePass := false, zeta := true, beta := true, eta := true, etaStruct := Lean.Meta.EtaStructMode.all, iota := true, proj := true, decide := false, arith := false, autoUnfold := false, dsimp := true, failIfUnchanged := true, ground := false, unfoldPartialApp := false, zetaDelta := false }) (simpOnly : optParam Bool false) (kind : optParam Lean.Elab.Tactic.SimpKind Lean.Elab.Tactic.SimpKind.simp) (dischargeWrapper : optParam Lean.Elab.Tactic.Simp.DischargeWrapper Lean.Elab.Tactic.Simp.DischargeWrapper.default) (hasStar : optParam Bool false) : Lean.MetaM Lean.Elab.Tactic.MkSimpContextResult

Make MkSimpContextResult giving data instead of Syntax. Doesn't support arguments. Intended to be very similar to Lean.Elab.Tactic.mkSimpContext Todo: support arguments.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Meta.mkSimpContext (cfg : optParam Lean.Meta.Simp.Config { maxSteps := Lean.Meta.Simp.defaultMaxSteps, maxDischargeDepth := 2, contextual := false, memoize := true, singlePass := false, zeta := true, beta := true, eta := true, etaStruct := Lean.Meta.EtaStructMode.all, iota := true, proj := true, decide := false, arith := false, autoUnfold := false, dsimp := true, failIfUnchanged := true, ground := false, unfoldPartialApp := false, zetaDelta := false }) (simpOnly : optParam Bool false) (kind : optParam Lean.Elab.Tactic.SimpKind Lean.Elab.Tactic.SimpKind.simp) (dischargeWrapper : optParam Lean.Elab.Tactic.Simp.DischargeWrapper Lean.Elab.Tactic.Simp.DischargeWrapper.default) (hasStar : optParam Bool false) : Lean.MetaM Lean.Meta.Simp.Context

Make Simp.Context giving data instead of Syntax. Doesn't support arguments. Intended to be very similar to Lean.Elab.Tactic.mkSimpContext Todo: support arguments.

Equations

• Lean.Meta.mkSimpContext cfg simpOnly kind dischargeWrapper hasStar = do let data ← Lean.Meta.mkSimpContextResult cfg simpOnly kind dischargeWrapper hasStar pure data.ctx

def hasSimpAttribute (env : Lean.Environment) (declName : Lean.Name) : Bool

Tests whether declName has the @[simp] attribute in env.

Equations

• hasSimpAttribute env declName = Lean.PersistentHashSet.contains (Lean.ScopedEnvExtension.getState Lean.Meta.simpExtension env).lemmaNames (Lean.Meta.Origin.decl declName true)

Declare notation classes.

def Lean.Parser.Attr.simpsArgsRest : Lean.ParserDescr

arguments to @[simps] attribute.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Parser.Attr.simps : Lean.ParserDescr

The @[simps] attribute automatically derives lemmas specifying the projections of this declaration.

Example:

@[simps] def foo : ℕ × ℤ := (1, 2)

derives two simp lemmas:

@[simp] lemma foo_fst : foo.fst = 1
@[simp] lemma foo_snd : foo.snd = 2

• It does not derive simp lemmas for the prop-valued projections.

• It will automatically reduce newly created beta-redexes, but will not unfold any definitions.

• If the structure has a coercion to either sorts or functions, and this is defined to be one of the projections, then this coercion will be used instead of the projection.

• If the structure is a class that has an instance to a notation class, like Neg or Mul, then this notation is used instead of the corresponding projection.

• You can specify custom projections, by giving a declaration with name {StructureName}.Simps.{projectionName}. See Note [custom simps projection].

  Example:

  def Equiv.Simps.invFun (e : α ≃ β) : β → α := e.symm
  @[simps] def Equiv.trans (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ :=
  ⟨e₂ ∘ e₁, e₁.symm ∘ e₂.symm⟩
  

  generates

  @[simp] lemma Equiv.trans_toFun : ∀ {α β γ} (e₁ e₂) (a : α), ⇑(e₁.trans e₂) a = (⇑e₂ ∘ ⇑e₁) a
  @[simp] lemma Equiv.trans_invFun : ∀ {α β γ} (e₁ e₂) (a : γ),
    ⇑((e₁.trans e₂).symm) a = (⇑(e₁.symm) ∘ ⇑(e₂.symm)) a
  

• You can specify custom projection names, by specifying the new projection names using initialize_simps_projections. Example: initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply). See initialize_simps_projections for more information.

• If one of the fields itself is a structure, this command will recursively create simp lemmas for all fields in that structure.

  • Exception: by default it will not recursively create simp lemmas for fields in the structures Prod, PProd, and Opposite. You can give explicit projection names or change the value of Simps.Config.notRecursive to override this behavior.

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_snd_fst : foo.snd.fst = 3
  @[simp] lemma foo_snd_snd : foo.snd.snd = 4
  

• You can use @[simps proj1 proj2 ...] to only generate the projection lemmas for the specified projections.

• Recursive projection names can be specified using proj1_proj2_proj3. This will create a lemma of the form foo.proj1.proj2.proj3 = ....

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps fst fst_fst snd] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_fst_fst : foo.fst.fst = 1
  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}
  

• If one of the values is an eta-expanded structure, we will eta-reduce this structure.

  Example:

  structure EquivPlusData (α β) extends α ≃ β where
    data : Bool
  @[simps] def EquivPlusData.rfl {α} : EquivPlusData α α := { Equiv.refl α with data := true }
  

  generates the following:

  @[simp] lemma bar_toEquiv : ∀ {α : Sort*}, bar.toEquiv = Equiv.refl α
  @[simp] lemma bar_data : ∀ {α : Sort*}, bar.data = true
  

  This is true, even though Lean inserts an eta-expanded version of Equiv.refl α in the definition of bar.

• For configuration options, see the doc string of Simps.Config.

• The precise syntax is simps (config := e)? ident*, where e : Expr is an expression of type Simps.Config and ident* is a list of desired projection names.

• @[simps] reduces let-expressions where necessary.

• When option trace.simps.verbose is true, simps will print the projections it finds and the lemmas it generates. The same can be achieved by using @[simps?].

• Use @[to_additive (attr := simps)] to apply both to_additive and simps to a definition This will also generate the additive versions of all simp lemmas.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Parser.Attr.attrSimps?_ : Lean.ParserDescr

The @[simps] attribute automatically derives lemmas specifying the projections of this declaration.

Example:

@[simps] def foo : ℕ × ℤ := (1, 2)

derives two simp lemmas:

@[simp] lemma foo_fst : foo.fst = 1
@[simp] lemma foo_snd : foo.snd = 2

• It does not derive simp lemmas for the prop-valued projections.

• It will automatically reduce newly created beta-redexes, but will not unfold any definitions.

• If the structure has a coercion to either sorts or functions, and this is defined to be one of the projections, then this coercion will be used instead of the projection.

• If the structure is a class that has an instance to a notation class, like Neg or Mul, then this notation is used instead of the corresponding projection.

• You can specify custom projections, by giving a declaration with name {StructureName}.Simps.{projectionName}. See Note [custom simps projection].

  Example:

  def Equiv.Simps.invFun (e : α ≃ β) : β → α := e.symm
  @[simps] def Equiv.trans (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ :=
  ⟨e₂ ∘ e₁, e₁.symm ∘ e₂.symm⟩
  

  generates

  @[simp] lemma Equiv.trans_toFun : ∀ {α β γ} (e₁ e₂) (a : α), ⇑(e₁.trans e₂) a = (⇑e₂ ∘ ⇑e₁) a
  @[simp] lemma Equiv.trans_invFun : ∀ {α β γ} (e₁ e₂) (a : γ),
    ⇑((e₁.trans e₂).symm) a = (⇑(e₁.symm) ∘ ⇑(e₂.symm)) a
  

• You can specify custom projection names, by specifying the new projection names using initialize_simps_projections. Example: initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply). See initialize_simps_projections for more information.

• If one of the fields itself is a structure, this command will recursively create simp lemmas for all fields in that structure.

  • Exception: by default it will not recursively create simp lemmas for fields in the structures Prod, PProd, and Opposite. You can give explicit projection names or change the value of Simps.Config.notRecursive to override this behavior.

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_snd_fst : foo.snd.fst = 3
  @[simp] lemma foo_snd_snd : foo.snd.snd = 4
  

• You can use @[simps proj1 proj2 ...] to only generate the projection lemmas for the specified projections.

• Recursive projection names can be specified using proj1_proj2_proj3. This will create a lemma of the form foo.proj1.proj2.proj3 = ....

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps fst fst_fst snd] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_fst_fst : foo.fst.fst = 1
  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}
  

• If one of the values is an eta-expanded structure, we will eta-reduce this structure.

  Example:

  structure EquivPlusData (α β) extends α ≃ β where
    data : Bool
  @[simps] def EquivPlusData.rfl {α} : EquivPlusData α α := { Equiv.refl α with data := true }
  

  generates the following:

  @[simp] lemma bar_toEquiv : ∀ {α : Sort*}, bar.toEquiv = Equiv.refl α
  @[simp] lemma bar_data : ∀ {α : Sort*}, bar.data = true
  

  This is true, even though Lean inserts an eta-expanded version of Equiv.refl α in the definition of bar.

• For configuration options, see the doc string of Simps.Config.

• The precise syntax is simps (config := e)? ident*, where e : Expr is an expression of type Simps.Config and ident* is a list of desired projection names.

• @[simps] reduces let-expressions where necessary.

• When option trace.simps.verbose is true, simps will print the projections it finds and the lemmas it generates. The same can be achieved by using @[simps?].

• Use @[to_additive (attr := simps)] to apply both to_additive and simps to a definition This will also generate the additive versions of all simp lemmas.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Parser.Attr.attrSimps!_ : Lean.ParserDescr

The @[simps] attribute automatically derives lemmas specifying the projections of this declaration.

Example:

@[simps] def foo : ℕ × ℤ := (1, 2)

derives two simp lemmas:

@[simp] lemma foo_fst : foo.fst = 1
@[simp] lemma foo_snd : foo.snd = 2

• It does not derive simp lemmas for the prop-valued projections.

• It will automatically reduce newly created beta-redexes, but will not unfold any definitions.

• If the structure has a coercion to either sorts or functions, and this is defined to be one of the projections, then this coercion will be used instead of the projection.

• If the structure is a class that has an instance to a notation class, like Neg or Mul, then this notation is used instead of the corresponding projection.

• You can specify custom projections, by giving a declaration with name {StructureName}.Simps.{projectionName}. See Note [custom simps projection].

  Example:

  def Equiv.Simps.invFun (e : α ≃ β) : β → α := e.symm
  @[simps] def Equiv.trans (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ :=
  ⟨e₂ ∘ e₁, e₁.symm ∘ e₂.symm⟩
  

  generates

  @[simp] lemma Equiv.trans_toFun : ∀ {α β γ} (e₁ e₂) (a : α), ⇑(e₁.trans e₂) a = (⇑e₂ ∘ ⇑e₁) a
  @[simp] lemma Equiv.trans_invFun : ∀ {α β γ} (e₁ e₂) (a : γ),
    ⇑((e₁.trans e₂).symm) a = (⇑(e₁.symm) ∘ ⇑(e₂.symm)) a
  

• You can specify custom projection names, by specifying the new projection names using initialize_simps_projections. Example: initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply). See initialize_simps_projections for more information.

• If one of the fields itself is a structure, this command will recursively create simp lemmas for all fields in that structure.

  • Exception: by default it will not recursively create simp lemmas for fields in the structures Prod, PProd, and Opposite. You can give explicit projection names or change the value of Simps.Config.notRecursive to override this behavior.

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_snd_fst : foo.snd.fst = 3
  @[simp] lemma foo_snd_snd : foo.snd.snd = 4
  

• You can use @[simps proj1 proj2 ...] to only generate the projection lemmas for the specified projections.

• Recursive projection names can be specified using proj1_proj2_proj3. This will create a lemma of the form foo.proj1.proj2.proj3 = ....

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps fst fst_fst snd] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_fst_fst : foo.fst.fst = 1
  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}
  

• If one of the values is an eta-expanded structure, we will eta-reduce this structure.

  Example:

  structure EquivPlusData (α β) extends α ≃ β where
    data : Bool
  @[simps] def EquivPlusData.rfl {α} : EquivPlusData α α := { Equiv.refl α with data := true }
  

  generates the following:

  @[simp] lemma bar_toEquiv : ∀ {α : Sort*}, bar.toEquiv = Equiv.refl α
  @[simp] lemma bar_data : ∀ {α : Sort*}, bar.data = true
  

  This is true, even though Lean inserts an eta-expanded version of Equiv.refl α in the definition of bar.

• For configuration options, see the doc string of Simps.Config.

• The precise syntax is simps (config := e)? ident*, where e : Expr is an expression of type Simps.Config and ident* is a list of desired projection names.

• @[simps] reduces let-expressions where necessary.

• When option trace.simps.verbose is true, simps will print the projections it finds and the lemmas it generates. The same can be achieved by using @[simps?].

• Use @[to_additive (attr := simps)] to apply both to_additive and simps to a definition This will also generate the additive versions of all simp lemmas.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Parser.Attr.attrSimps!?_ : Lean.ParserDescr

The @[simps] attribute automatically derives lemmas specifying the projections of this declaration.

Example:

@[simps] def foo : ℕ × ℤ := (1, 2)

derives two simp lemmas:

@[simp] lemma foo_fst : foo.fst = 1
@[simp] lemma foo_snd : foo.snd = 2

• It does not derive simp lemmas for the prop-valued projections.

• It will automatically reduce newly created beta-redexes, but will not unfold any definitions.

• If the structure has a coercion to either sorts or functions, and this is defined to be one of the projections, then this coercion will be used instead of the projection.

• If the structure is a class that has an instance to a notation class, like Neg or Mul, then this notation is used instead of the corresponding projection.

• You can specify custom projections, by giving a declaration with name {StructureName}.Simps.{projectionName}. See Note [custom simps projection].

  Example:

  def Equiv.Simps.invFun (e : α ≃ β) : β → α := e.symm
  @[simps] def Equiv.trans (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ :=
  ⟨e₂ ∘ e₁, e₁.symm ∘ e₂.symm⟩
  

  generates

  @[simp] lemma Equiv.trans_toFun : ∀ {α β γ} (e₁ e₂) (a : α), ⇑(e₁.trans e₂) a = (⇑e₂ ∘ ⇑e₁) a
  @[simp] lemma Equiv.trans_invFun : ∀ {α β γ} (e₁ e₂) (a : γ),
    ⇑((e₁.trans e₂).symm) a = (⇑(e₁.symm) ∘ ⇑(e₂.symm)) a
  

• You can specify custom projection names, by specifying the new projection names using initialize_simps_projections. Example: initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply). See initialize_simps_projections for more information.

• If one of the fields itself is a structure, this command will recursively create simp lemmas for all fields in that structure.

  • Exception: by default it will not recursively create simp lemmas for fields in the structures Prod, PProd, and Opposite. You can give explicit projection names or change the value of Simps.Config.notRecursive to override this behavior.

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_snd_fst : foo.snd.fst = 3
  @[simp] lemma foo_snd_snd : foo.snd.snd = 4
  

• You can use @[simps proj1 proj2 ...] to only generate the projection lemmas for the specified projections.

• Recursive projection names can be specified using proj1_proj2_proj3. This will create a lemma of the form foo.proj1.proj2.proj3 = ....

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps fst fst_fst snd] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_fst_fst : foo.fst.fst = 1
  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}
  

• If one of the values is an eta-expanded structure, we will eta-reduce this structure.

  Example:

  structure EquivPlusData (α β) extends α ≃ β where
    data : Bool
  @[simps] def EquivPlusData.rfl {α} : EquivPlusData α α := { Equiv.refl α with data := true }
  

  generates the following:

  @[simp] lemma bar_toEquiv : ∀ {α : Sort*}, bar.toEquiv = Equiv.refl α
  @[simp] lemma bar_data : ∀ {α : Sort*}, bar.data = true
  

  This is true, even though Lean inserts an eta-expanded version of Equiv.refl α in the definition of bar.

• For configuration options, see the doc string of Simps.Config.

• The precise syntax is simps (config := e)? ident*, where e : Expr is an expression of type Simps.Config and ident* is a list of desired projection names.

• @[simps] reduces let-expressions where necessary.

• When option trace.simps.verbose is true, simps will print the projections it finds and the lemmas it generates. The same can be achieved by using @[simps?].

• Use @[to_additive (attr := simps)] to apply both to_additive and simps to a definition This will also generate the additive versions of all simp lemmas.

Equations

• One or more equations did not get rendered due to their size.

def Lean.Parser.Attr.attrSimps?!_ : Lean.ParserDescr

The @[simps] attribute automatically derives lemmas specifying the projections of this declaration.

Example:

@[simps] def foo : ℕ × ℤ := (1, 2)

derives two simp lemmas:

@[simp] lemma foo_fst : foo.fst = 1
@[simp] lemma foo_snd : foo.snd = 2

• It does not derive simp lemmas for the prop-valued projections.

• It will automatically reduce newly created beta-redexes, but will not unfold any definitions.

• If the structure has a coercion to either sorts or functions, and this is defined to be one of the projections, then this coercion will be used instead of the projection.

• If the structure is a class that has an instance to a notation class, like Neg or Mul, then this notation is used instead of the corresponding projection.

• You can specify custom projections, by giving a declaration with name {StructureName}.Simps.{projectionName}. See Note [custom simps projection].

  Example:

  def Equiv.Simps.invFun (e : α ≃ β) : β → α := e.symm
  @[simps] def Equiv.trans (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ :=
  ⟨e₂ ∘ e₁, e₁.symm ∘ e₂.symm⟩
  

  generates

  @[simp] lemma Equiv.trans_toFun : ∀ {α β γ} (e₁ e₂) (a : α), ⇑(e₁.trans e₂) a = (⇑e₂ ∘ ⇑e₁) a
  @[simp] lemma Equiv.trans_invFun : ∀ {α β γ} (e₁ e₂) (a : γ),
    ⇑((e₁.trans e₂).symm) a = (⇑(e₁.symm) ∘ ⇑(e₂.symm)) a
  

• You can specify custom projection names, by specifying the new projection names using initialize_simps_projections. Example: initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply). See initialize_simps_projections for more information.

• If one of the fields itself is a structure, this command will recursively create simp lemmas for all fields in that structure.

  • Exception: by default it will not recursively create simp lemmas for fields in the structures Prod, PProd, and Opposite. You can give explicit projection names or change the value of Simps.Config.notRecursive to override this behavior.

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_snd_fst : foo.snd.fst = 3
  @[simp] lemma foo_snd_snd : foo.snd.snd = 4
  

• You can use @[simps proj1 proj2 ...] to only generate the projection lemmas for the specified projections.

• Recursive projection names can be specified using proj1_proj2_proj3. This will create a lemma of the form foo.proj1.proj2.proj3 = ....

  Example:

  structure MyProd (α β : Type*) := (fst : α) (snd : β)
  @[simps fst fst_fst snd] def foo : Prod ℕ ℕ × MyProd ℕ ℕ := ⟨⟨1, 2⟩, 3, 4⟩
  

  generates

  @[simp] lemma foo_fst : foo.fst = (1, 2)
  @[simp] lemma foo_fst_fst : foo.fst.fst = 1
  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}
  

• If one of the values is an eta-expanded structure, we will eta-reduce this structure.

  Example:

  structure EquivPlusData (α β) extends α ≃ β where
    data : Bool
  @[simps] def EquivPlusData.rfl {α} : EquivPlusData α α := { Equiv.refl α with data := true }
  

  generates the following:

  @[simp] lemma bar_toEquiv : ∀ {α : Sort*}, bar.toEquiv = Equiv.refl α
  @[simp] lemma bar_data : ∀ {α : Sort*}, bar.data = true
  

  This is true, even though Lean inserts an eta-expanded version of Equiv.refl α in the definition of bar.

• For configuration options, see the doc string of Simps.Config.

• The precise syntax is simps (config := e)? ident*, where e : Expr is an expression of type Simps.Config and ident* is a list of desired projection names.

• @[simps] reduces let-expressions where necessary.

• When option trace.simps.verbose is true, simps will print the projections it finds and the lemmas it generates. The same can be achieved by using @[simps?].

• Use @[to_additive (attr := simps)] to apply both to_additive and simps to a definition This will also generate the additive versions of all simp lemmas.

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opaque Lean.Parser.linter.simpsNoConstructor : Lean.Option Bool

Linter to check that simps! is used when needed

opaque Lean.Parser.linter.simpsUnusedCustomDeclarations : Lean.Option Bool

Linter to check that no unused custom declarations are declared for simps.

def Lean.Parser.Command.simpsRule.rename : Lean.ParserDescr

Syntax for renaming a projection in initialize_simps_projections.

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def Lean.Parser.Command.simpsRule.erase : Lean.ParserDescr

Syntax for making a projection non-default in initialize_simps_projections.

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def Lean.Parser.Command.simpsRule.add : Lean.ParserDescr

Syntax for making a projection default in initialize_simps_projections.

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def Lean.Parser.Command.simpsRule.prefix : Lean.ParserDescr

Syntax for making a projection prefix.

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def Lean.Parser.Command.simpsRule : Lean.ParserDescr

Syntax for a single rule in initialize_simps_projections.

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def Lean.Parser.Command.simpsProj : Lean.ParserDescr

Syntax for initialize_simps_projections.

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def Lean.Parser.Command.initialize_simps_projections : Lean.ParserDescr

This command specifies custom names and custom projections for the simp attribute simpsAttr.

• You can specify custom names by writing e.g. initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply).
• See Note [custom simps projection] and the examples below for information how to declare custom projections.
• For algebraic structures, we will automatically use the notation (like Mul) for the projections if such an instance is available.
• By default, the projections to parent structures are not default projections, but all the data-carrying fields are (including those in parent structures).
• You can disable a projection by default by running initialize_simps_projections Equiv (-invFun) This will ensure that no simp lemmas are generated for this projection, unless this projection is explicitly specified by the user.
• Conversely, you can enable a projection by default by running initialize_simps_projections Equiv (+toEquiv).
• If you want the projection name added as a prefix in the generated lemma name, you can use as_prefix fieldName: initialize_simps_projections Equiv (toFun → coe, as_prefix coe) Note that this does not influence the parsing of projection names: if you have a declaration foo and you want to apply the projections snd, coe (which is a prefix) and fst, in that order you can run @[simps snd_coe_fst] def foo ... and this will generate a lemma with the name coe_foo_snd_fst.
  • Run initialize_simps_projections? (or set_option trace.simps.verbose true) to see the generated projections.
• Running initialize_simps_projections MyStruc without arguments is not necessary, it has the same effect if you just add @[simps] to a declaration.
• It is recommended to call @[simps] or initialize_simps_projections in the same file as the structure declaration. Otherwise, the projections could be generated multiple times in different files.

Some common uses:

• If you define a new homomorphism-like structure (like MulHom) you can just run initialize_simps_projections after defining the DFunLike instance (or instance that implies a DFunLike instance).

    instance {mM : Mul M} {mN : Mul N} : DFunLike (MulHom M N) M N := ...
    initialize_simps_projections MulHom (toFun → apply)
  

  This will generate foo_apply lemmas for each declaration foo.
• If you prefer coe_foo lemmas that state equalities between functions, use initialize_simps_projections MulHom (toFun → coe, as_prefix coe) In this case you have to use @[simps (config := .asFn)] or equivalently @[simps (config := .asFn)] whenever you call @[simps].
• You can also initialize to use both, in which case you have to choose which one to use by default, by using either of the following

    initialize_simps_projections MulHom (toFun → apply, toFun → coe, as_prefix coe, -coe)
    initialize_simps_projections MulHom (toFun → apply, toFun → coe, as_prefix coe, -apply)
  

  In the first case, you can get both lemmas using @[simps, simps (config := .asFn) coe] and in the second case you can get both lemmas using @[simps (config := .asFn), simps apply].
• If you declare a new homomorphism-like structure (like RelEmbedding), then initialize_simps_projections will automatically find any DFunLike coercions that will be used as the default projection for the toFun field.

    initialize_simps_projections relEmbedding (toFun → apply)
  

• If you have an isomorphism-like structure (like Equiv) you often want to define a custom projection for the inverse:

    def Equiv.Simps.symm_apply (e : α ≃ β) : β → α := e.symm
    initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply)
  

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def Lean.Parser.Command.commandInitialize_simps_projections?_ : Lean.ParserDescr

This command specifies custom names and custom projections for the simp attribute simpsAttr.

• You can specify custom names by writing e.g. initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply).
• See Note [custom simps projection] and the examples below for information how to declare custom projections.
• For algebraic structures, we will automatically use the notation (like Mul) for the projections if such an instance is available.
• By default, the projections to parent structures are not default projections, but all the data-carrying fields are (including those in parent structures).
• You can disable a projection by default by running initialize_simps_projections Equiv (-invFun) This will ensure that no simp lemmas are generated for this projection, unless this projection is explicitly specified by the user.
• Conversely, you can enable a projection by default by running initialize_simps_projections Equiv (+toEquiv).
• If you want the projection name added as a prefix in the generated lemma name, you can use as_prefix fieldName: initialize_simps_projections Equiv (toFun → coe, as_prefix coe) Note that this does not influence the parsing of projection names: if you have a declaration foo and you want to apply the projections snd, coe (which is a prefix) and fst, in that order you can run @[simps snd_coe_fst] def foo ... and this will generate a lemma with the name coe_foo_snd_fst.
  • Run initialize_simps_projections? (or set_option trace.simps.verbose true) to see the generated projections.
• Running initialize_simps_projections MyStruc without arguments is not necessary, it has the same effect if you just add @[simps] to a declaration.
• It is recommended to call @[simps] or initialize_simps_projections in the same file as the structure declaration. Otherwise, the projections could be generated multiple times in different files.

Some common uses:

• If you define a new homomorphism-like structure (like MulHom) you can just run initialize_simps_projections after defining the DFunLike instance (or instance that implies a DFunLike instance).

    instance {mM : Mul M} {mN : Mul N} : DFunLike (MulHom M N) M N := ...
    initialize_simps_projections MulHom (toFun → apply)
  

  This will generate foo_apply lemmas for each declaration foo.
• If you prefer coe_foo lemmas that state equalities between functions, use initialize_simps_projections MulHom (toFun → coe, as_prefix coe) In this case you have to use @[simps (config := .asFn)] or equivalently @[simps (config := .asFn)] whenever you call @[simps].
• You can also initialize to use both, in which case you have to choose which one to use by default, by using either of the following

    initialize_simps_projections MulHom (toFun → apply, toFun → coe, as_prefix coe, -coe)
    initialize_simps_projections MulHom (toFun → apply, toFun → coe, as_prefix coe, -apply)
  

  In the first case, you can get both lemmas using @[simps, simps (config := .asFn) coe] and in the second case you can get both lemmas using @[simps (config := .asFn), simps apply].
• If you declare a new homomorphism-like structure (like RelEmbedding), then initialize_simps_projections will automatically find any DFunLike coercions that will be used as the default projection for the toFun field.

    initialize_simps_projections relEmbedding (toFun → apply)
  

• If you have an isomorphism-like structure (like Equiv) you often want to define a custom projection for the inverse:

    def Equiv.Simps.symm_apply (e : α ≃ β) : β → α := e.symm
    initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply)
  

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structure Simps.ProjectionData : Type

Projection data for a single projection of a structure

• name : Lean.Name

  The name used in the generated simp lemmas

• expr : Lean.Expr

  An Expression used by simps for the projection. It must be definitionally equal to an original projection (or a composition of multiple projections). These Expressions can contain the universe parameters specified in the first argument of structureExt.

• projNrs : List Nat

  A list of natural numbers, which is the projection number(s) that have to be applied to the Expression. For example the list [0, 1] corresponds to applying the first projection of the structure, and then the second projection of the resulting structure (this assumes that the target of the first projection is a structure with at least two projections). The composition of these projections is required to be definitionally equal to the provided Expression.

• isDefault : Bool

  A boolean specifying whether simp lemmas are generated for this projection by default.

• isPrefix : Bool

  A boolean specifying whether this projection is written as prefix.

instance Simps.instInhabitedProjectionData : Inhabited Simps.ProjectionData

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• Simps.instInhabitedProjectionData = { default := { name := default, expr := default, projNrs := default, isDefault := default, isPrefix := default } }

instance Simps.instToMessageDataProjectionData : Lean.ToMessageData Simps.ProjectionData

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opaque Simps.structureExt : Lean.NameMapExtension (List Lean.Name × Array Simps.ProjectionData)

The Simps.structureExt environment extension specifies the preferred projections of the given structure, used by the @[simps] attribute.

• You can generate this with the command initialize_simps_projections.
• If not generated, the @[simps] attribute will generate this automatically.
• To change the default value, see Note [custom simps projection].
• The first argument is the list of names of the universe variables used in the structure
• The second argument is an array that consists of the projection data for each projection.

structure Simps.ParsedProjectionData : Type

Projection data used internally in getRawProjections.

• strName : Lean.Name

  name for this projection used in the structure definition

• strStx : Lean.Syntax

  syntax that might have provided strName

• newName : Lean.Name

  name for this projection used in the generated simp lemmas

• newStx : Lean.Syntax

  syntax that provided newName

• isDefault : Bool

  will simp lemmas be generated for with (without specifically naming this?)

• isPrefix : Bool

  is the projection name a prefix?

• expr? : Option Lean.Expr

  projection expression

• projNrs : Array Nat

  the list of projection numbers this expression corresponds to

• isCustom : Bool

  is this a projection that is changed by the user?

def Simps.ParsedProjectionData.toProjectionData (p : Simps.ParsedProjectionData) : Simps.ProjectionData

Turn ParsedProjectionData into ProjectionData.

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• Simps.ParsedProjectionData.toProjectionData p = { name := p.newName, expr := Option.getD p.expr? default, projNrs := Array.toList p.projNrs, isDefault := p.isDefault, isPrefix := p.isPrefix }

instance Simps.instToMessageDataParsedProjectionData : Lean.ToMessageData Simps.ParsedProjectionData

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inductive Simps.ProjectionRule : Type

The type of rules that specify how metadata for projections in changes. See initialize_simps_projections.

• rename: Lean.Name → Lean.Syntax → Lean.Name → Lean.Syntax → Simps.ProjectionRule

  A renaming rule before→after or Each name comes with the syntax used to write the rule, which is used to declare hover information.

• add: Lean.Name → Lean.Syntax → Simps.ProjectionRule

  An adding rule +fieldName

• erase: Lean.Name → Lean.Syntax → Simps.ProjectionRule

  A hiding rule -fieldName

• prefix: Lean.Name → Lean.Syntax → Simps.ProjectionRule

  A prefix rule prefix fieldName

instance Simps.instToMessageDataProjectionRule : Lean.ToMessageData Simps.ProjectionRule

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def Simps.projectionsInfo (l : List Simps.ProjectionData) (pref : String) (str : Lean.Name) : Lean.MessageData

Returns the projection information of a structure.

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def Simps.findProjectionIndices (strName : Lean.Name) (projName : Lean.Name) : Lean.MetaM (List Nat)

Find the indices of the projections that need to be applied to elaborate $e.$projName. Example: If e : α ≃+ β and projName = `invFun then this returns [0, 1], because the first projection of MulEquiv is toEquiv and the second projection of Equiv is invFun.

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partial def Simps.getCompositeOfProjectionsAux (proj : String) (e : Lean.Expr) (pos : Array Nat) (args : Array Lean.Expr) : Lean.MetaM (Lean.Expr × Array Nat)

Auxiliary function of getCompositeOfProjections.

def Simps.getCompositeOfProjections (structName : Lean.Name) (proj : String) : Lean.MetaM (Lean.Expr × Array Nat)

Suppose we are given a structure str and a projection proj, that could be multiple nested projections (separated by _), where each projection could be a projection of a parent structure. This function returns an expression that is the composition of these projections and a list of natural numbers, that are the projection numbers of the applied projections. Note that this function is similar to elaborating dot notation, but it can do a little more. Example: if we do

structure gradedFun (A : ℕ → Type*) where
  toFun := ∀ i j, A i →+ A j →+ A (i + j)
initialize_simps_projections (toFun_toFun_toFun → myMul)

we will be able to generate the "projection" λ {A} (f : gradedFun A) (x : A i) (y : A j) ↦ ↑(↑(f.toFun i j) x) y, which projection notation cannot do.

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def Simps.mkParsedProjectionData (structName : Lean.Name) : Lean.CoreM (Array Simps.ParsedProjectionData)

Get the default ParsedProjectionData for structure str. It first returns the direct fields of the structure in the right order, and then all (non-subobject fields) of all parent structures. The subobject fields are precisely the non-default fields.

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def Simps.applyProjectionRules (projs : Array Simps.ParsedProjectionData) (rules : Array Simps.ProjectionRule) : Lean.CoreM (Array Simps.ParsedProjectionData)

Execute the projection renamings (and turning off projections) as specified by rules.

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def Simps.findProjection (str : Lean.Name) (proj : Simps.ParsedProjectionData) (rawUnivs : List Lean.Level) : Lean.CoreM Simps.ParsedProjectionData

Auxiliary function for getRawProjections. Generates the default projection, and looks for a custom projection declared by the user, and replaces the default projection with the custom one, if it can find it.

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def Simps.checkForUnusedCustomProjs (stx : Lean.Syntax) (str : Lean.Name) (projs : Array Simps.ParsedProjectionData) : Lean.CoreM Unit

Checks if there are declarations in the current file in the namespace {str}.Simps that are not used.

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def Simps.findAutomaticProjectionsAux (str : Lean.Name) (proj : Simps.ParsedProjectionData) (args : Array Lean.Expr) : Lean.Elab.TermElabM (Option (Lean.Expr × Lean.Name))

If a structure has a field that corresponds to a coercion to functions or sets, or corresponds to notation, find the custom projection that uses this coercion or notation. Returns the custom projection and the name of the projection used.

We catch most errors this function causes, so that we don't fail if an unrelated projection has an applicable name. (e.g. Iso.inv)

Implementation note: getting rid of TermElabM is tricky, since Expr.mkAppOptM doesn't allow to keep metavariables around, which are necessary for OutParam.

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def Simps.findAutomaticProjections (str : Lean.Name) (projs : Array Simps.ParsedProjectionData) : Lean.CoreM (Array Simps.ParsedProjectionData)

Auxiliary function for getRawProjections. Find custom projections, automatically found by simps. These come from DFunLike and SetLike instances.

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def Simps.getRawProjections (stx : Lean.Syntax) (str : Lean.Name) (traceIfExists : optParam Bool false) (rules : optParam (Array Simps.ProjectionRule) #[]) (trc : optParam Bool false) : Lean.CoreM (List Lean.Name × Array Simps.ProjectionData)

Get the projections used by simps associated to a given structure str.

The returned information is also stored in the environment extension Simps.structureExt, which is given to str. If str already has this attribute, the information is read from this extension instead. See the documentation for this extension for the data this tactic returns.

The returned universe levels are the universe levels of the structure. For the projections there are three cases

• If the declaration {StructureName}.Simps.{projectionName} has been declared, then the value of this declaration is used (after checking that it is definitionally equal to the actual projection. If you rename the projection name, the declaration should have the new projection name.
• You can also declare a custom projection that is a composite of multiple projections.
• Otherwise, for every class with the notation_class attribute, and the structure has an instance of that notation class, then the projection of that notation class is used for the projection that is definitionally equal to it (if there is such a projection). This means in practice that coercions to function types and sorts will be used instead of a projection, if this coercion is definitionally equal to a projection. Furthermore, for notation classes like Mul and Zero those projections are used instead of the corresponding projection. Projections for coercions and notation classes are not automatically generated if they are composites of multiple projections (for example when you use extend without the oldStructureCmd (does this exist?)).
• Otherwise, the projection of the structure is chosen. For example: getRawProjections env `Prod gives the default projections.

  ([u, v], [(`fst, `(Prod.fst.{u v}), [0], true, false),
     (`snd, `(@Prod.snd.{u v}), [1], true, false)])

Optionally, this command accepts three optional arguments:

• If traceIfExists the command will always generate a trace message when the structure already has an entry in structureExt.
• The rules argument specifies whether projections should be added, renamed, used as prefix, and not used by default.
• if trc is true, this tactic will trace information just as if set_option trace.simps.verbose true was set.

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def Simps.elabSimpsRule : Lean.Syntax → Lean.Elab.Command.CommandElabM Simps.ProjectionRule

Parse a rule for initialize_simps_projections. It is <name>→<name>, -<name>, +<name> or as_prefix <name>.

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def Simps.elabInitializeSimpsProjections : Lean.Elab.Command.CommandElab

Function elaborating initialize_simps_projections.

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structure Simps.Config : Type

Configuration options for @[simps]

• isSimp : Bool

  Make generated lemmas simp lemmas

• attrs : List Lean.Name

  Other simp-attributes to apply to generated lemmas. Attributes that are currently not simp-attributes are not supported.

• simpRhs : Bool

  simplify the right-hand side of generated simp-lemmas using dsimp, simp.

• typeMd : Lean.Meta.TransparencyMode

  TransparencyMode used to reduce the type in order to detect whether it is a structure.

• rhsMd : Lean.Meta.TransparencyMode

  TransparencyMode used to reduce the right-hand side in order to detect whether it is a constructor. Note: was none in Lean 3

• fullyApplied : Bool

  Generated lemmas that are fully applied, i.e. generates equalities between applied functions. Set this to false to generate equalities between functions.

• notRecursive : List Lean.Name

  List of types in which we are not recursing to generate simplification lemmas. E.g. if we write @[simps] def e : α × β ≃ β × α := ... we will generate e_apply and not e_apply_fst.

• debug : Bool

  Output debug messages. Not used much, use set_option simps.debug true instead.

instance Simps.instInhabitedConfig : Inhabited Simps.Config

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def Simps.elabSimpsConfig : Lean.Syntax → Lean.Elab.TermElabM Simps.Config

Function elaborating Config

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def Simps.Config.asFn : Simps.Config

A common configuration for @[simps]: generate equalities between functions instead equalities between fully applied Expressions. Use this using @[simps (config := .asFn)].

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def Simps.Config.lemmasOnly : Simps.Config

A common configuration for @[simps]: don't tag the generated lemmas with @[simp]. Use this using @[simps (config := .lemmasOnly)].

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def Lean.Expr.instantiateLambdasOrApps (es : Array Lean.Expr) (e : Lean.Expr) : Lean.Expr

instantiateLambdasOrApps es e instantiates lambdas in e by expressions from es. If the length of es is larger than the number of lambdas in e, then the term is applied to the remaining terms. Also reduces head let-expressions in e, including those after instantiating all lambdas.

This is very similar to expr.substs, but this also reduces head let-expressions.

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• Lean.Expr.instantiateLambdasOrApps es e = Lean.Expr.betaRev e (Array.reverse es) true

def Simps.getProjectionExprs (stx : Lean.Syntax) (tgt : Lean.Expr) (rhs : Lean.Expr) (cfg : Simps.Config) : Lean.MetaM (Array (Lean.Expr × Simps.ProjectionData))

Get the projections of a structure used by @[simps] applied to the appropriate arguments. Returns a list of tuples

(corresponding right-hand-side, given projection name, projection Expression,
  future projection numbers, used by default, is prefix)

(where all fields except the first are packed in a ProjectionData structure) one for each projection. The given projection name is the name for the projection used by the user used to generate (and parse) projection names. For example, in the structure

Example 1: getProjectionExprs env `(α × β) `(⟨x, y⟩) will give the output

  [(`(x), `fst, `(@Prod.fst.{u v} α β), [], true, false),
   (`(y), `snd, `(@Prod.snd.{u v} α β), [], true, false)]

Example 2: getProjectionExprs env `(α ≃ α) `(⟨id, id, fun _ ↦ rfl, fun _ ↦ rfl⟩) will give the output

  [(`(id), `apply, (Equiv.toFun), [], true, false),
   (`(id), `symm_apply, (fun e ↦ e.symm.toFun), [], true, false),
   ...,
   ...]

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def Simps.addProjection (ref : Lean.Syntax) (univs : List Lean.Name) (declName : Lean.Name) (type : Lean.Expr) (lhs : Lean.Expr) (rhs : Lean.Expr) (args : Array Lean.Expr) (cfg : Simps.Config) : Lean.MetaM Unit

Add a lemma with nm stating that lhs = rhs. type is the type of both lhs and rhs, args is the list of local constants occurring, and univs is the list of universe variables.

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partial def Simps.headStructureEtaReduce (e : Lean.Expr) : Lean.MetaM Lean.Expr

Perform head-structure-eta-reduction on expression e. That is, if e is of the form ⟨f.1, f.2, ..., f.n⟩ with f definitionally equal to e, then headStructureEtaReduce e = headStructureEtaReduce f and headStructureEtaReduce e = e otherwise.

partial def Simps.addProjections (ref : Lean.Syntax) (univs : List Lean.Name) (nm : Lean.Name) (type : Lean.Expr) (lhs : Lean.Expr) (rhs : Lean.Expr) (args : Array Lean.Expr) (mustBeStr : Bool) (cfg : Simps.Config) (todo : List (String × Lean.Syntax)) (toApply : List Nat) : Lean.MetaM (Array Lean.Name)

Derive lemmas specifying the projections of the declaration. nm: name of the lemma If todo is non-empty, it will generate exactly the names in todo. toApply is non-empty after a custom projection that is a composition of multiple projections was just used. In that case we need to apply these projections before we continue changing lhs. simpLemmas: names of the simp lemmas added so far.(simpLemmas : Array Name)

def simpsTac (ref : Lean.Syntax) (nm : Lean.Name) (cfg : optParam Simps.Config { isSimp := true, attrs := [], simpRhs := false, typeMd := Lean.Meta.TransparencyMode.instances, rhsMd := Lean.Meta.TransparencyMode.reducible, fullyApplied := true, notRecursive := [`Prod, `PProd, `Opposite, `PreOpposite], debug := false }) (todo : optParam (List (String × Lean.Syntax)) []) (trc : optParam Bool false) : Lean.AttrM (Array Lean.Name)

simpsTac derives simp lemmas for all (nested) non-Prop projections of the declaration. If todo is non-empty, it will generate exactly the names in todo. If shortNm is true, the generated names will only use the last projection name. If trc is true, trace as if trace.simps.verbose is true.

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def simpsTacFromSyntax (nm : Lean.Name) (stx : Lean.Syntax) : Lean.AttrM (Array Lean.Name)

elaborate the syntax and run simpsTac.

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opaque simpsAttr : Lean.ParametricAttribute (Array Lean.Name)

The simps attribute.