Loogle!

Result

Found 2152 declarations mentioning Equiv, Equiv.symm, and Eq. Of these, 78 have a name containing ".symm_". Of these, 14 match your pattern(s).

• Equiv.symm_symm 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u} {β : Sort v} (e : α ≃ β) : e.symm.symm = e
• Equiv.symm_eq_iff_trans_eq_refl 📋 Mathlib.Logic.Equiv.Defs
  {α : Sort u} {β : Sort v} {f : α ≃ β} {g : β ≃ α} : f.symm = g ↔ f.trans g = Equiv.refl α
• Equiv.symm_swap 📋 Mathlib.Logic.Equiv.Basic
  {α : Sort u_1} [DecidableEq α] (a b : α) : Equiv.symm (Equiv.swap a b) = Equiv.swap a b
• Function.Involutive.symm_eq_self_of_involutive 📋 Mathlib.Logic.Equiv.Basic
  {α : Sort u_1} (f : Equiv.Perm α) (h : Function.Involutive ⇑f) : Equiv.symm f = f
• AddEquiv.symm_addMonoidHomCongrLeftEquiv 📋 Mathlib.Algebra.Group.Equiv.Basic
  {M₁ : Type u_5} {M₂ : Type u_6} {N : Type u_8} [AddZeroClass M₁] [AddZeroClass M₂] [AddMonoid N] (e : M₁ ≃+ M₂) : e.addMonoidHomCongrLeftEquiv.symm = e.symm.addMonoidHomCongrLeftEquiv
• MulEquiv.symm_monoidHomCongrLeftEquiv 📋 Mathlib.Algebra.Group.Equiv.Basic
  {M₁ : Type u_5} {M₂ : Type u_6} {N : Type u_8} [MulOneClass M₁] [MulOneClass M₂] [Monoid N] (e : M₁ ≃* M₂) : e.monoidHomCongrLeftEquiv.symm = e.symm.monoidHomCongrLeftEquiv
• AddEquiv.symm_addMonoidHomCongrRightEquiv 📋 Mathlib.Algebra.Group.Equiv.Basic
  {M : Type u_4} {N₁ : Type u_9} {N₂ : Type u_10} [AddZeroClass M] [AddMonoid N₁] [AddMonoid N₂] (e : N₁ ≃+ N₂) : e.addMonoidHomCongrRightEquiv.symm = e.symm.addMonoidHomCongrRightEquiv
• MulEquiv.symm_monoidHomCongrRightEquiv 📋 Mathlib.Algebra.Group.Equiv.Basic
  {M : Type u_4} {N₁ : Type u_9} {N₂ : Type u_10} [MulOneClass M] [Monoid N₁] [Monoid N₂] (e : N₁ ≃* N₂) : e.monoidHomCongrRightEquiv.symm = e.symm.monoidHomCongrRightEquiv
• Equiv.symm_divLeft 📋 Mathlib.Algebra.Group.Units.Equiv
  {G : Type u_5} [CommGroup G] (a : G) : (Equiv.divLeft a).symm = Equiv.divLeft a
• Equiv.symm_subLeft 📋 Mathlib.Algebra.Group.Units.Equiv
  {G : Type u_5} [AddCommGroup G] (a : G) : (Equiv.subLeft a).symm = Equiv.subLeft a
• Flag.symm_map 📋 Mathlib.Order.Preorder.Chain
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) : (Flag.map e).symm = Flag.map e.symm
• AlgEquiv.symm_toEquiv_eq_symm 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂] {e : A₁ ≃ₐ[R] A₂} : (↑e).symm = ↑e.symm
• Equiv.symm_simpleGraph 📋 Mathlib.Combinatorics.SimpleGraph.Maps
  {V : Type u_1} {W : Type u_2} (e : V ≃ W) : e.simpleGraph.symm = e.symm.simpleGraph
• DFA.symm_reindex 📋 Mathlib.Computability.DFA
  {α : Type u} {σ : Type v} {σ' : Type u_2} (g : σ ≃ σ') : (DFA.reindex g).symm = DFA.reindex g.symm

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 6ff4759 serving mathlib revision 519f454