Loogle!

Result

Found 33 declarations mentioning Preorder.toLT, Cardinal, PartialOrder.toPreorder, Cardinal.mk, Cardinal.aleph0, Cardinal.partialOrder, and LT.lt. Of these, 6 match your pattern(s).

• Cardinal.aleph0_lt_mk 📋 Mathlib.SetTheory.Cardinal.Basic
  {α : Type u} [Uncountable α] : Cardinal.aleph0 < Cardinal.mk α
• Cardinal.aleph0_lt_mk_iff 📋 Mathlib.SetTheory.Cardinal.Basic
  {α : Type u} : Cardinal.aleph0 < Cardinal.mk α ↔ Uncountable α
• IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt' 📋 Mathlib.FieldTheory.IsAlgClosed.Classification
  {R : Type u} [CommRing R] {K' : Type u} [Field K'] [Algebra R K'] [IsAlgClosed K'] {ι' : Type u} (v' : ι' → K') [Nontrivial R] (hv : IsTranscendenceBasis R v') (hR : Cardinal.mk R ≤ Cardinal.aleph0) (hK : Cardinal.aleph0 < Cardinal.mk K') : Cardinal.mk K' = Cardinal.mk ι'
• IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt 📋 Mathlib.FieldTheory.IsAlgClosed.Classification
  {R : Type u} {K : Type v} [CommRing R] [Field K] [Algebra R K] [IsAlgClosed K] {ι : Type w} (v : ι → K) [Nontrivial R] (hv : IsTranscendenceBasis R v) (hR : Cardinal.mk R ≤ Cardinal.aleph0) (hK : Cardinal.aleph0 < Cardinal.mk K) : Cardinal.lift.{w, v} (Cardinal.mk K) = Cardinal.lift.{v, w} (Cardinal.mk ι)
• IsAlgClosed.ringEquiv_of_equiv_of_charZero 📋 Mathlib.FieldTheory.IsAlgClosed.Classification
  {K : Type u} {L : Type v} [Field K] [Field L] [IsAlgClosed K] [IsAlgClosed L] [CharZero K] [CharZero L] (hK : Cardinal.aleph0 < Cardinal.mk K) (hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L)
• IsAlgClosed.ringEquiv_of_equiv_of_char_eq 📋 Mathlib.FieldTheory.IsAlgClosed.Classification
  {K : Type u} {L : Type v} [Field K] [Field L] [IsAlgClosed K] [IsAlgClosed L] (p : ℕ) [CharP K p] [CharP L p] (hK : Cardinal.aleph0 < Cardinal.mk K) (hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L)

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 401c76f serving mathlib revision b87935d