Loogle!

Result

Found 3765 declarations mentioning Prod.fst and Eq. Of these, 107 have a name containing "_fst_". Of these, 28 match your pattern(s).

• Std.DHashMap.Internal.Raw.containsThenInsertIfNew_fst_eq 📋 Std.Data.DHashMap.Internal.Raw
  {α : Type u} {β : α → Type v} [BEq α] [Hashable α] {m : Std.DHashMap.Raw α β} (h : m.WF) {a : α} {b : β a} : (m.containsThenInsertIfNew a b).1 = (Std.DHashMap.Internal.Raw₀.containsThenInsertIfNew ⟨m, ⋯⟩ a b).1
• Std.DHashMap.Internal.Raw.containsThenInsert_fst_eq 📋 Std.Data.DHashMap.Internal.Raw
  {α : Type u} {β : α → Type v} [BEq α] [Hashable α] {m : Std.DHashMap.Raw α β} (h : m.WF) {a : α} {b : β a} : (m.containsThenInsert a b).1 = (Std.DHashMap.Internal.Raw₀.containsThenInsert ⟨m, ⋯⟩ a b).1
• Std.DHashMap.Internal.Raw.Const.getThenInsertIfNew?_fst_eq 📋 Std.Data.DHashMap.Internal.Raw
  {α : Type u} {β : Type v} [BEq α] [Hashable α] {m : Std.DHashMap.Raw α fun x => β} (h : m.WF) {a : α} {b : β} : (Std.DHashMap.Raw.Const.getThenInsertIfNew? m a b).1 = (Std.DHashMap.Internal.Raw₀.Const.getThenInsertIfNew? ⟨m, ⋯⟩ a b).1
• Std.DHashMap.Internal.Raw.getThenInsertIfNew?_fst_eq 📋 Std.Data.DHashMap.Internal.Raw
  {α : Type u} {β : α → Type v} [BEq α] [Hashable α] [LawfulBEq α] {m : Std.DHashMap.Raw α β} (h : m.WF) {a : α} {b : β a} : (m.getThenInsertIfNew? a b).1 = (Std.DHashMap.Internal.Raw₀.getThenInsertIfNew? ⟨m, ⋯⟩ a b).1
• Std.DTreeMap.Internal.Impl.containsThenInsertIfNew!_fst_eq_containsₘ 📋 Std.Data.DTreeMap.Internal.Model
  {α : Type u} {β : α → Type v} [Ord α] [Std.TransOrd α] (t : Std.DTreeMap.Internal.Impl α β) (a : α) (b : β a) : (Std.DTreeMap.Internal.Impl.containsThenInsertIfNew! a b t).1 = t.containsₘ a
• Std.DTreeMap.Internal.Impl.containsThenInsertIfNew_fst_eq_containsₘ 📋 Std.Data.DTreeMap.Internal.Model
  {α : Type u} {β : α → Type v} [Ord α] [Std.TransOrd α] (t : Std.DTreeMap.Internal.Impl α β) (htb : t.Balanced) (a : α) (b : β a) : (Std.DTreeMap.Internal.Impl.containsThenInsertIfNew a b t htb).1 = t.containsₘ a
• Std.DTreeMap.Internal.Impl.containsThenInsert!_fst_eq_containsThenInsert_fst 📋 Std.Data.DTreeMap.Internal.Model
  {α : Type u} {β : α → Type v} [Ord α] (t : Std.DTreeMap.Internal.Impl α β) (htb : t.Balanced) (a : α) (b : β a) : (Std.DTreeMap.Internal.Impl.containsThenInsert! a b t).1 = (Std.DTreeMap.Internal.Impl.containsThenInsert a b t htb).1
• Std.DTreeMap.Internal.Impl.containsThenInsertIfNew!_fst_eq_containsThenInsertIfNew_fst 📋 Std.Data.DTreeMap.Internal.Model
  {α : Type u} {β : α → Type v} [Ord α] (t : Std.DTreeMap.Internal.Impl α β) (htb : t.Balanced) (a : α) (b : β a) : (Std.DTreeMap.Internal.Impl.containsThenInsertIfNew! a b t).1 = (Std.DTreeMap.Internal.Impl.containsThenInsertIfNew a b t htb).1
• Std.DTreeMap.Internal.Impl.containsThenInsert_fst_eq_containsₘ 📋 Std.Data.DTreeMap.Internal.WF.Lemmas
  {α : Type u} {β : α → Type v} [Ord α] [Std.TransOrd α] [BEq α] [Std.LawfulBEqOrd α] (t : Std.DTreeMap.Internal.Impl α β) (htb : t.Balanced) (ho : t.Ordered) (a : α) (b : β a) : (Std.DTreeMap.Internal.Impl.containsThenInsert a b t htb).1 = t.containsₘ a
• Std.Sat.AIG.relabelNat'_fst_eq_relabelNat 📋 Std.Sat.AIG.RelabelNat
  {α : Type} [DecidableEq α] [Hashable α] {aig : Std.Sat.AIG α} : aig.relabelNat'.1 = aig.relabelNat
• Prod.eq_iff_fst_eq_snd_eq 📋 Mathlib.Data.Prod.Basic
  {α : Type u_1} {β : Type u_2} {p q : α × β} : p = q ↔ p.1 = q.1 ∧ p.2 = q.2
• Finset.filter_fst_eq_antidiagonal 📋 Mathlib.Algebra.Order.Antidiag.Prod
  {A : Type u_1} [AddCommMonoid A] [PartialOrder A] [CanonicallyOrderedAdd A] [Sub A] [OrderedSub A] [AddLeftReflectLE A] [Finset.HasAntidiagonal A] (n m : A) [DecidablePred fun x => x = m] [Decidable (m ≤ n)] : {x ∈ Finset.antidiagonal n | x.1 = m} = if m ≤ n then {(m, n - m)} else ∅
• Submodule.prodEquivOfIsCompl_symm_apply_fst_eq_zero 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] (p q : Submodule R E) (h : IsCompl p q) {x : E} : ((p.prodEquivOfIsCompl q h).symm x).1 = 0 ↔ x ∈ q
• CategoryTheory.Limits.Types.binaryProductIso_hom_comp_fst_apply 📋 Mathlib.CategoryTheory.Limits.Types.Products
  (X Y : Type u) (x : X ⨯ Y) : ((CategoryTheory.Limits.Types.binaryProductIso X Y).hom x).1 = CategoryTheory.Limits.prod.fst x
• GenContFract.IntFractPair.seq1_fst_eq_of 📋 Mathlib.Algebra.ContinuedFractions.Computation.Translations
  {K : Type u_1} [DivisionRing K] [LinearOrder K] [FloorRing K] {v : K} : (GenContFract.IntFractPair.seq1 v).1 = GenContFract.IntFractPair.of v
• Subgroup.IsComplement.equiv_fst_eq_self_of_mem_of_one_mem 📋 Mathlib.GroupTheory.Complement
  {G : Type u_1} [Group G] {S T : Set G} (hST : Subgroup.IsComplement S T) {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) : (hST.equiv g).1 = ⟨g, hg⟩
• Subgroup.IsComplement.equiv_fst_eq_one_of_mem_of_one_mem 📋 Mathlib.GroupTheory.Complement
  {G : Type u_1} [Group G] {S T : Set G} (hST : Subgroup.IsComplement S T) {g : G} (h1 : 1 ∈ S) (hg : g ∈ T) : (hST.equiv g).1 = ⟨1, h1⟩
• Subgroup.IsComplement.equiv_fst_eq_iff_leftCosetEquivalence 📋 Mathlib.GroupTheory.Complement
  {G : Type u_1} [Group G] {K : Subgroup G} {S : Set G} (hSK : Subgroup.IsComplement S ↑K) {g₁ g₂ : G} : (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ LeftCosetEquivalence (↑K) g₁ g₂
• Set.AddAntidiagonal.eq_of_fst_eq_fst 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [AddCommMonoid α] [IsCancelAdd α] {s t : Set α} {a : α} {x y : ↑(s.addAntidiagonal t a)} (h : (↑x).1 = (↑y).1) : x = y
• Set.MulAntidiagonal.eq_of_fst_eq_fst 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [CommMonoid α] [IsCancelMul α] {s t : Set α} {a : α} {x y : ↑(s.mulAntidiagonal t a)} (h : (↑x).1 = (↑y).1) : x = y
• Set.AddAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [AddCommMonoid α] [IsCancelAdd α] {s t : Set α} {a : α} {x y : ↑(s.addAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [CommMonoid α] [IsCancelMul α] {s t : Set α} {a : α} {x y : ↑(s.mulAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• Set.SMulAntidiagonal.eq_of_fst_eq_fst 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [SMul G P] [IsCancelSMul G P] {x y : ↑(s.smulAntidiagonal t a)} (h : (↑x).1 = (↑y).1) : x = y
• Set.VAddAntidiagonal.eq_of_fst_eq_fst 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [VAdd G P] [IsCancelVAdd G P] {x y : ↑(s.vaddAntidiagonal t a)} (h : (↑x).1 = (↑y).1) : x = y
• Set.SMulAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [SMul G P] [IsCancelSMul G P] {x y : ↑(s.smulAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• Set.VAddAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [VAdd G P] [IsCancelVAdd G P] {x y : ↑(s.vaddAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• SimpleGraph.dart_fst_fiber_card_eq_degree 📋 Mathlib.Combinatorics.SimpleGraph.DegreeSum
  {V : Type u} (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] [DecidableEq V] (v : V) : {d | d.toProd.1 = v}.card = G.degree v
• SimpleGraph.dart_fst_fiber 📋 Mathlib.Combinatorics.SimpleGraph.DegreeSum
  {V : Type u} (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj] [DecidableEq V] (v : V) : {d | d.toProd.1 = v} = Finset.image (G.dartOfNeighborSet v) Finset.univ

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 ee6d0f1