name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
RelSeries.last_drop | Mathlib.Order.RelSeries | ∀ {α : Type u_1} {r : SetRel α α} (p : RelSeries r) (i : Fin (p.length + 1)), (p.drop i).last = p.last | null | true |
CategoryTheory.MorphismProperty.Arrow.isoMk_inv_left | Mathlib.CategoryTheory.MorphismProperty.Comma | ∀ {T : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} T] {P Q W : CategoryTheory.MorphismProperty T}
[inst_1 : Q.IsMultiplicative] [inst_2 : W.IsMultiplicative] [inst_3 : Q.RespectsIso] [inst_4 : W.RespectsIso]
{A B : P.Arrow Q W} (f : A.left ≅ B.left) (g : A.right ≅ B.right)
(w :
autoParam (CategoryThe... | null | true |
Function.locallyFinsuppWithin.addSubgroup._proof_1 | Mathlib.Topology.LocallyFinsupp | ∀ {X : Type u_1} [inst : TopologicalSpace X] (U : Set X) {Y : Type u_2} [inst_1 : AddGroup Y] {f : X → Y},
f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ nhds z, (t ∩ Function.support f).Finite} →
-f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ nhds z, (t ∩ Function.support f).Finite} | null | false |
Lean.Lsp.TextDocumentEdit.ctorIdx | Lean.Data.Lsp.Basic | Lean.Lsp.TextDocumentEdit → ℕ | null | false |
Vector.instDecidableExistsVectorZero | Init.Data.Vector.Lemmas | {α : Type u_1} → (P : Vector α 0 → Prop) → [Decidable (P #v[])] → Decidable (∃ xs, P xs) | null | true |
_private.Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits.0.SimpleGraph.ErdosStone.filter_subset_compl_verts | Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | ∀ {n : ℕ} {G : SimpleGraph (Fin n)} [inst : DecidableRel G.Adj] {r t t' : ℕ} (K : G.CompleteEquipartiteSubgraph r t'),
SimpleGraph.ErdosStone.filter✝ K t ⊆ K.vertsᶜ | null | true |
AddCommGrpCat.hasColimit_of_small_quot | Mathlib.Algebra.Category.Grp.Colimits | ∀ {J : Type u} [inst : CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J AddCommGrpCat)
[inst_1 : DecidableEq J], Small.{w, max u w} (AddCommGrpCat.Colimits.Quot F) → CategoryTheory.Limits.HasColimit F | null | true |
_private.Mathlib.Topology.Inseparable.0.inseparable_prod._simp_1_2 | Mathlib.Topology.Inseparable | ∀ {α : Type u_1} {β : Type u_2} {f₁ f₂ : Filter α} {g₁ g₂ : Filter β} [f₁.NeBot] [g₁.NeBot],
(f₁ ×ˢ g₁ = f₂ ×ˢ g₂) = (f₁ = f₂ ∧ g₁ = g₂) | null | false |
CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_X₁ | Mathlib.Algebra.Homology.SpectralObject.Page | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ)
(hn₁ : autoParam (n₀ + 1 = n₁) Categ... | null | true |
Subsemigroup.comap_top | Mathlib.Algebra.Group.Subsemigroup.Operations | ∀ {M : Type u_1} {N : Type u_2} [inst : Mul M] [inst_1 : Mul N] (f : M →ₙ* N), Subsemigroup.comap f ⊤ = ⊤ | null | true |
MeasureTheory.SimpleFunc.instNonAssocSemiring._proof_4 | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [inst_1 : NonAssocSemiring β],
autoParam (∀ (n : ℕ), ↑(n + 1) = ↑n + 1) AddMonoidWithOne.natCast_succ._autoParam | null | false |
_private.Lean.Server.CodeActions.Basic.0.Lean.Server.handleCodeActionResolve.match_3 | Lean.Server.CodeActions.Basic | (motive : Option (IO Lean.Lsp.CodeAction) → Sort u_1) →
(x : Option (IO Lean.Lsp.CodeAction)) →
((lazy : IO Lean.Lsp.CodeAction) → motive (some lazy)) →
((x : Option (IO Lean.Lsp.CodeAction)) → motive x) → motive x | null | false |
_private.Mathlib.Util.AddRelatedDecl.0.Mathlib.Tactic.checkImplicitTransparency._sparseCasesOn_1 | Mathlib.Util.AddRelatedDecl | {motive : Lean.ReducibilityStatus → Sort u} →
(t : Lean.ReducibilityStatus) →
motive Lean.ReducibilityStatus.semireducible → (Nat.hasNotBit 2 t.ctorIdx → motive t) → motive t | null | false |
_private.Lean.Meta.Constructions.CtorElim.0.Lean.initFn._regBuiltin._private.Lean.Meta.Constructions.CtorElim.0.Lean.initFn.docString_1._@.Lean.Meta.Constructions.CtorElim.299025572._hygCtx._hyg.2 | Lean.Meta.Constructions.CtorElim | IO Unit | null | false |
_private.Mathlib.RingTheory.Polynomial.Bernstein.0.bernsteinPolynomial.variance._simp_1_2 | Mathlib.RingTheory.Polynomial.Bernstein | ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c | null | false |
Lean.Syntax.TSepArray.mk | Init.Prelude | {ks : Lean.SyntaxNodeKinds} → {sep : String} → Array Lean.Syntax → Lean.Syntax.TSepArray ks sep | null | true |
Lean.Int.mkInstHPow | Lean.Expr | Lean.Expr | null | true |
normalize_eq_zero._simp_1 | Mathlib.Algebra.GCDMonoid.Basic | ∀ {α : Type u_1} [inst : MonoidWithZero α] [inst_1 : NormalizationMonoid α] {x : α}, (normalize x = 0) = (x = 0) | null | false |
CategoryTheory.Functor.mapCoconeWhisker_hom_hom | Mathlib.CategoryTheory.Limits.Cones | ∀ {J : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} K]
{C : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄}
[inst_3 : CategoryTheory.Category.{v₄, u₄} D] (H : CategoryTheory.Functor C D) {F : CategoryTheory.Functor J C}
{E : Cat... | null | true |
ContMDiffSection.toFun | Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {𝕜 : Type u_1} →
[inst : NontriviallyNormedField 𝕜] →
{E : Type u_2} →
[inst_1 : NormedAddCommGroup E] →
[inst_2 : NormedSpace 𝕜 E] →
{H : Type u_3} →
[inst_3 : TopologicalSpace H] →
{I : ModelWithCorners 𝕜 E H} →
{M : Type u_4} →
... | the underlying function of this section | true |
_private.Mathlib.RingTheory.HahnSeries.Summable.0.HahnSeries.SummableFamily.hsum._simp_1 | Mathlib.RingTheory.HahnSeries.Summable | ∀ {Γ : Type u_1} {R : Type u_3} [inst : PartialOrder Γ] [inst_1 : Zero R] (x : HahnSeries Γ R) (a : Γ),
(a ∈ x.support) = (x.coeff a ≠ 0) | null | false |
Finsupp.mapRange.equiv._proof_2 | Mathlib.Data.Finsupp.Defs | ∀ {ι : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Zero M] [inst_1 : Zero N] (e : M ≃ N) (hf : e 0 = 0)
(x : ι →₀ M), Finsupp.mapRange ⇑e.symm ⋯ (Finsupp.mapRange (⇑e) hf x) = x | null | false |
WithTop.instMulZeroOneClass._proof_2 | Mathlib.Algebra.Order.Ring.WithTop | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : MulZeroOneClass α] [Nontrivial α] (x : WithTop α), x * 1 = x | null | false |
Std.TreeMap.getKey_insert | Std.Data.TreeMap.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap α β cmp} [inst : Std.TransCmp cmp] {k a : α}
{v : β} {h₁ : a ∈ t.insert k v}, (t.insert k v).getKey a h₁ = if h₂ : cmp k a = Ordering.eq then k else t.getKey a ⋯ | null | true |
Std.Http.Header.Name.setCookie | Std.Http.Data.Headers.Name | Std.Http.Header.Name | Standard Set-Cookie header name (server → client)
| true |
Polynomial.orderOf_root_cyclotomic_dvd | Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | ∀ {n : ℕ} (hpos : 0 < n) {p : ℕ} [inst : Fact (Nat.Prime p)] {a : ℕ}
(hroot : (Polynomial.cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)),
orderOf (ZMod.unitOfCoprime a ⋯) ∣ n | If `(a : ℕ)` is a root of `cyclotomic n (ZMod p)`, then the multiplicative order of `a` modulo
`p` divides `n`. | true |
alternatingGroup.normal_subgroup_eq_bot_or_eq_top_of_card_ne_six | Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | ∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : Fintype α],
5 ≤ Nat.card α → Nat.card α ≠ 6 → ∀ {N : Subgroup ↥(alternatingGroup α)} [N.Normal], N = ⊥ ∨ N = ⊤ | If `α` has at least 5 elements, but not 6,
then the only nontrivial normal subgroup of `alternatingGroup α`
is `⊤`. | true |
Equiv.setCongr | Mathlib.Logic.Equiv.Set | {α : Type u_3} → {β : Type u_4} → α ≃ β → Set α ≃ Set β | If `α` is equivalent to `β`, then `Set α` is equivalent to `Set β`.
See also `Equiv.finsetCongr`. | true |
_private.Mathlib.GroupTheory.SpecificGroups.Dihedral.0.instDecidableEqDihedralGroup.decEq.match_1.eq_4 | Mathlib.GroupTheory.SpecificGroups.Dihedral | ∀ {n : ℕ} (motive : DihedralGroup n → DihedralGroup n → Sort u_1) (a b : ZMod n)
(h_1 : (a b : ZMod n) → motive (DihedralGroup.r a) (DihedralGroup.r b))
(h_2 : (a a_1 : ZMod n) → motive (DihedralGroup.r a) (DihedralGroup.sr a_1))
(h_3 : (a a_1 : ZMod n) → motive (DihedralGroup.sr a) (DihedralGroup.r a_1))
(h_4 ... | null | true |
_private.Mathlib.SetTheory.Cardinal.Cofinality.Ordinal.0.Ordinal.exists_ord_cof_eq._simp_1_2 | Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | ∀ {α : Sort u_1} {p : α → Prop} {a b : Subtype p}, (↑a = ↑b) = (a = b) | null | false |
Submodule.toLocalized' | Mathlib.Algebra.Module.LocalizedModule.Submodule | {R : Type u_1} →
(S : Type u_2) →
{M : Type u_3} →
{N : Type u_4} →
[inst : CommSemiring R] →
[inst_1 : CommSemiring S] →
[inst_2 : AddCommMonoid M] →
[inst_3 : AddCommMonoid N] →
[inst_4 : Module R M] →
[inst_5 : Module R N] →
... | The localization map of a submodule. | true |
_private.Mathlib.Computability.EpsilonNFA.0.εNFA.MemεClosure.base | Mathlib.Computability.EpsilonNFA | ∀ {α : Type u} {σ : Type v} {M : εNFA α σ} {S : Set σ}, ∀ s ∈ S, εNFA.MemεClosure✝ M S s | null | true |
Nat.getD_toList_ric_eq_fallback | Init.Data.Range.Polymorphic.NatLemmas | ∀ {n i fallback : ℕ}, n < i → (*...=n).toList.getD i fallback = fallback | null | true |
_private.Mathlib.Topology.UniformSpace.Compact.0.IsClosed.relPreimage_of_isCompact._simp_1_1 | Mathlib.Topology.UniformSpace.Compact | ∀ {α : Type u} (s : Set α) (x : α), (x ∈ sᶜ) = (x ∉ s) | null | false |
SimpleGraph.neighborFinset_nonempty._simp_1 | Mathlib.Combinatorics.SimpleGraph.Finite | ∀ {V : Type u_1} (G : SimpleGraph V) (v : V) [inst : Fintype ↑(G.neighborSet v)],
(G.neighborFinset v).Nonempty = ¬G.IsIsolated v | null | false |
isGLB_ciInf_set | Mathlib.Order.ConditionallyCompleteLattice.Indexed | ∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompleteLattice α] {f : β → α} {s : Set β},
BddBelow (f '' s) → s.Nonempty → IsGLB (f '' s) (⨅ i, f ↑i) | null | true |
SSet.instFiniteObjOppositeSimplexCategoryTensorObj | Mathlib.AlgebraicTopology.SimplicialSet.Monoidal | ∀ (X Y : SSet) (n : SimplexCategoryᵒᵖ) [Finite (X.obj n)] [Finite (Y.obj n)],
Finite ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj n) | null | true |
Fin.one_eq_zero_iff | Init.Data.Fin.Lemmas | ∀ {n : ℕ} [inst : NeZero n], 1 = 0 ↔ n = 1 | null | true |
Finset.attach_empty | Mathlib.Data.Finset.Basic | ∀ {α : Type u_1}, ∅.attach = ∅ | null | true |
antitone_toDual_comp_iff | Mathlib.Order.Monotone.Basic | ∀ {α : Type u} {β : Type v} [inst : Preorder α] [inst_1 : Preorder β] {f : α → β},
Antitone (⇑OrderDual.toDual ∘ f) ↔ Monotone f | null | true |
MulEquiv.piMultiplicative._proof_2 | Mathlib.Algebra.Group.Equiv.TypeTags | ∀ {ι : Type u_1} (K : ι → Type u_2),
Function.RightInverse (fun x => Multiplicative.ofAdd fun i => Multiplicative.toAdd (x i)) fun x i =>
Multiplicative.ofAdd (Multiplicative.toAdd x i) | null | false |
_private.Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.0.Pi.opensMeasurableSpace_of_subsingleton._simp_2 | Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | ∀ {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {s : Set α},
IsOpen s = (s ∈ Set.preimage f '' {s | IsOpen s}) | null | false |
_private.Init.Data.BitVec.Lemmas.0.BitVec.shiftLeft_eq_concat_of_lt._proof_1_1 | Init.Data.BitVec.Lemmas | ∀ {w n : ℕ}, ∀ i < w, ¬i < n → ¬i - n < w → False | null | false |
ModelWithCorners.MfldCat.Hom._mkInternal.injEq | Mathlib.Geometry.Manifold.Category.MfldCat.OfModel | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {n : WithTop ℕ∞}
{M N : I.MfldCat n} (hom' hom'_1 : ContMDiffMap I I (↑M) (↑N) n),
({ hom' := hom' } = { hom' ... | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.filter_equiv_self_iff._simp_1_4 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {β : α → Type v} {t t' : Std.DTreeMap.Internal.Impl α β}, t.Equiv t' = t.toListModel.Perm t'.toListModel | null | false |
Finset.exists_subsuperset_card_eq | Mathlib.Data.Finset.Card | ∀ {α : Type u_1} {s t : Finset α} {n : ℕ}, s ⊆ t → s.card ≤ n → n ≤ t.card → ∃ u, s ⊆ u ∧ u ⊆ t ∧ u.card = n | Given a subset `s` of a set `t`, of sizes at most and at least `n` respectively, there exists a
set `u` of size `n` which is both a superset of `s` and a subset of `t`. | true |
ContDiffMapSupportedIn._sizeOf_inst | Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | (E : Type u_2) →
(F : Type u_3) →
{inst : NormedAddCommGroup E} →
{inst_1 : NormedSpace ℝ E} →
{inst_2 : NormedAddCommGroup F} →
{inst_3 : NormedSpace ℝ F} →
(n : ℕ∞) →
(K : TopologicalSpace.Compacts E) → [SizeOf E] → [SizeOf F] → SizeOf (ContDiffMapSupportedIn E ... | null | false |
SheafOfModules.mk.noConfusion | Mathlib.Algebra.Category.ModuleCat.Sheaf | {C : Type u₁} →
{inst : CategoryTheory.Category.{v₁, u₁} C} →
{J : CategoryTheory.GrothendieckTopology C} →
{R : CategoryTheory.Sheaf J RingCat} →
{P : Sort u_1} →
{val : PresheafOfModules R.obj} →
{isSheaf : CategoryTheory.Presheaf.IsSheaf J val.presheaf} →
{val'... | null | false |
_private.Aesop.Util.EqualUpToIds.0.Aesop.EqualUpToIds.Unsafe.exprsEqualUpToIdsCore₃._sparseCasesOn_3 | Aesop.Util.EqualUpToIds | {motive : Lean.Expr → Sort u} →
(t : Lean.Expr) →
((deBruijnIndex : ℕ) → motive (Lean.Expr.bvar deBruijnIndex)) →
((data : Lean.MData) → (expr : Lean.Expr) → motive (Lean.Expr.mdata data expr)) →
((mvarId : Lean.MVarId) → motive (Lean.Expr.mvar mvarId)) → (Nat.hasNotBit 1029 t.ctorIdx → motive t) → ... | null | false |
Orientation.rotation_pi_div_two | Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | ∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : Fact (Module.finrank ℝ V = 2)]
(o : Orientation ℝ V (Fin 2)), o.rotation ↑(Real.pi / 2) = o.rightAngleRotation | Rotation by π / 2 is the "right-angle-rotation" map `J`. | true |
MeasureTheory.Measure.isAddLeftInvariant_eq_smul_of_innerRegular | Mathlib.MeasureTheory.Measure.Haar.Unique | ∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : IsTopologicalAddGroup G]
[inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G)
[inst_6 : μ.IsAddHaarMeasure] [inst_7 : MeasureTheory.IsFiniteMeasureOnCompacts μ'] [inst_8 : μ'.IsAddLe... | **Uniqueness of left-invariant measures**:
Given two left-invariant measures which are finite
on compacts and inner regular, they coincide up to a multiplicative constant. | true |
Lean.Meta.Grind.Arith.Linear.IneqCnstrProof.ring.elim | Lean.Meta.Tactic.Grind.Arith.Linear.Types | {motive_4 : Lean.Meta.Grind.Arith.Linear.IneqCnstrProof → Sort u} →
(t : Lean.Meta.Grind.Arith.Linear.IneqCnstrProof) →
t.ctorIdx = 2 →
((c : Lean.Meta.Grind.Arith.Linear.RingIneqCnstr) →
(lhs : Lean.Meta.Grind.Arith.Linear.LinExpr) →
motive_4 (Lean.Meta.Grind.Arith.Linear.IneqCnstrPro... | null | false |
String.posGE_eq_posGE_toSlice | Init.Data.String.Lemmas.FindPos | ∀ {s : String} {p : String.Pos.Raw} (h : p ≤ s.rawEndPos), s.posGE p h = String.Pos.ofToSlice (s.toSlice.posGE p ⋯) | null | true |
LaurentPolynomial.trunc | Mathlib.Algebra.Polynomial.Laurent | {R : Type u_1} → [inst : Semiring R] → LaurentPolynomial R →+ Polynomial R | `trunc : R[T;T⁻¹] →+ R[X]` maps a Laurent polynomial `f` to the polynomial whose terms of
nonnegative degree coincide with the ones of `f`. The terms of negative degree of `f` "vanish".
`trunc` is a left-inverse to `Polynomial.toLaurent`. | true |
CategoryTheory.Oplax.OplaxTrans.Hom.noConfusion | Mathlib.CategoryTheory.Bicategory.Modification.Oplax | {P : Sort u} →
{B : Type u₁} →
{inst : CategoryTheory.Bicategory B} →
{C : Type u₂} →
{inst_1 : CategoryTheory.Bicategory C} →
{F G : CategoryTheory.OplaxFunctor B C} →
{η θ : F ⟶ G} →
{t : CategoryTheory.Oplax.OplaxTrans.Hom η θ} →
{B' : Type u₁} ... | null | false |
RingCon.instNonAssocCommRingQuotient | Mathlib.RingTheory.Congruence.Defs | {R : Type u_1} → [inst : NonAssocCommRing R] → (c : RingCon R) → NonAssocCommRing c.Quotient | null | true |
_private.Mathlib.Order.Filter.FilterProduct.0.Filter.Germ.coe_lt._simp_1_4 | Mathlib.Order.Filter.FilterProduct | ∀ {α : Type u} {f : Ultrafilter α} {p : α → Prop}, (∀ᶠ (x : α) in ↑f, ¬p x) = ¬∀ᶠ (x : α) in ↑f, p x | null | false |
Rep.indCoindIso | Mathlib.RepresentationTheory.FiniteIndex | {k : Type u} →
{G : Type v} →
[inst : CommRing k] →
[inst_1 : Group G] →
{S : Subgroup G} →
[DecidableRel ⇑(QuotientGroup.rightRel S)] →
[S.FiniteIndex] →
(A : Rep.{max w u, u, v} k ↥S) → Rep.ind S.subtype A ≅ Rep.coind.{u, v, v, max u w} S.subtype A | Let `S ≤ G` be a finite index subgroup, `g₁, ..., gₙ` a set of right coset representatives of
`S`, and `A` a `k`-linear `S`-representation. This is an isomorphism `Ind_S^G(A) ≅ Coind_S^G(A)`.
The forward map sends `(⟦g ⊗ₜ[k] a⟧, sg) ↦ ρ(s)(a)`, and the inverse sends `f : G → A` to
`∑ᵢ ⟦gᵢ ⊗ₜ[k] f(gᵢ)⟧` for `1 ≤ i ≤ n`.... | true |
CategoryTheory.mateEquiv_conjugateEquiv_vcomp | Mathlib.CategoryTheory.Adjunction.Mates | ∀ {A : Type u₁} {B : Type u₂} {C : Type u₃} {D : Type u₄} [inst : CategoryTheory.Category.{v₁, u₁} A]
[inst_1 : CategoryTheory.Category.{v₂, u₂} B] [inst_2 : CategoryTheory.Category.{v₃, u₃} C]
[inst_3 : CategoryTheory.Category.{v₄, u₄} D] {G : CategoryTheory.Functor A C} {H : CategoryTheory.Functor B D}
{L₁ : Ca... | The mates equivalence commutes with this composition, essentially by `mateEquiv_vcomp`. | true |
NumberField.mixedEmbedding.unitSMul_smul | Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | ∀ (K : Type u_1) [inst : Field K] (u : (NumberField.RingOfIntegers K)ˣ) (x : NumberField.mixedEmbedding.mixedSpace K),
u • x = (NumberField.mixedEmbedding K) ((algebraMap (NumberField.RingOfIntegers K) K) ↑u) * x | null | true |
_private.Mathlib.Combinatorics.SetFamily.AhlswedeZhang.0.Finset.sups_aux | Mathlib.Combinatorics.SetFamily.AhlswedeZhang | ∀ {α : Type u_1} [inst : DistribLattice α] [inst_1 : DecidableEq α] {s t : Finset α} {a : α},
a ∈ upperClosure ↑(s ⊻ t) ↔ a ∈ upperClosure ↑s ∧ a ∈ upperClosure ↑t | null | true |
BitVec.smod_eq | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} (x y : BitVec w),
x.smod y =
match x.msb, y.msb with
| false, false => x.umod y
| false, true =>
have u := x.umod (-y);
if u = 0#w then u else u + y
| true, false =>
have u := (-x).umod y;
if u = 0#w then u else y - u
| true, true => -(-x).umod (-y) | Equation theorem for `smod` in terms of `umod`. | true |
List.reverseAux_reverseAux_nil | Init.Data.List.Lemmas | ∀ {α : Type u_1} {as bs : List α}, (as.reverseAux bs).reverseAux [] = bs.reverseAux as | null | true |
spectrum.subset_subalgebra | Mathlib.Algebra.Algebra.Spectrum.Basic | ∀ {S : Type u_1} {R : Type u_2} {A : Type u_3} [inst : CommSemiring R] [inst_1 : Ring A] [inst_2 : Algebra R A]
[inst_3 : SetLike S A] [inst_4 : SubringClass S A] [inst_5 : SMulMemClass S R A] {s : S} (a : ↥s),
spectrum R ↑a ⊆ spectrum R a | null | true |
IsPrimitiveRoot.isIntegral | Mathlib.RingTheory.RootsOfUnity.Minpoly | ∀ {n : ℕ} {K : Type u_1} [inst : CommRing K] {μ : K}, IsPrimitiveRoot μ n → 0 < n → IsIntegral ℤ μ | `μ` is integral over `ℤ`. | true |
Rep.instAdditiveResFunctor | Mathlib.RepresentationTheory.Rep.Res | ∀ {k : Type u} [inst : Semiring k] {G : Type v1} {H : Type v2} [inst_1 : Monoid G] [inst_2 : Monoid H] (f : H →* G),
(Rep.resFunctor f).Additive | null | true |
Equiv.nonUnitalCommSemiring._proof_1 | Mathlib.Algebra.Ring.TransferInstance | ∀ {α : Type u_2} {β : Type u_1} (e : α ≃ β) [inst : NonUnitalCommSemiring β], e (e.symm 0) = 0 | null | false |
ContinuousAt.fun_nsmul | Mathlib.Topology.Algebra.Monoid | ∀ {M : Type u_3} {X : Type u_5} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace M] [inst_2 : AddMonoid M]
[ContinuousAdd M] {f : X → M} {x : X}, ContinuousAt f x → ∀ (n : ℕ), ContinuousAt (fun i => n • f i) x | null | true |
QPF.fixToW._proof_1 | Mathlib.Data.QPF.Univariate.Basic | ∀ {F : Type u_1 → Type u_1} [q : QPF F] (x y : (QPF.P F).W),
QPF.Wequiv x y → QPF.recF (fun x => PFunctor.W.mk (QPF.repr x)) x = QPF.recF (fun x => PFunctor.W.mk (QPF.repr x)) y | null | false |
FiberBundleCore.localTrivAsPartialEquiv_target | Mathlib.Topology.FiberBundle.Basic | ∀ {ι : Type u_1} {B : Type u_2} {F : Type u_3} [inst : TopologicalSpace B] [inst_1 : TopologicalSpace F]
(Z : FiberBundleCore ι B F) (i : ι), (Z.localTrivAsPartialEquiv i).target = (Z.localTriv i).target | null | true |
_private.Mathlib.Geometry.Manifold.ContMDiff.Constructions.0.contMDiffWithinAt_pi_space._simp_1_4 | Mathlib.Geometry.Manifold.ContMDiff.Constructions | ∀ {α : Sort u_1} {p q : α → Prop}, (∀ (x : α), p x ∧ q x) = ((∀ (x : α), p x) ∧ ∀ (x : α), q x) | null | false |
Std.TreeSet.size_toArray | Std.Data.TreeSet.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {t : Std.TreeSet α cmp} [Std.TransCmp cmp], t.toArray.size = t.size | null | true |
_private.Mathlib.CategoryTheory.Filtered.Basic.0.CategoryTheory.IsFiltered.cocone_nonempty._simp_1_4 | Mathlib.CategoryTheory.Filtered.Basic | ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, b ∧ p x) = (b ∧ ∃ x, p x) | null | false |
_private.Mathlib.Geometry.Euclidean.Angle.Incenter.0.Affine.Triangle.oangle_excenter_singleton_eq._proof_1_4 | Mathlib.Geometry.Euclidean.Angle.Incenter | ∀ {V : Type u_2} {P : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
[inst_3 : NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₃ : Fin 3},
i₁ ≠ i₃ → ∃ x, ¬x = i₃ ∧ t.points x = t.points i₁ | null | false |
Lean.Meta.Grind.EMatchTheoremConstraint.guard.injEq | Lean.Meta.Tactic.Grind.Extension | ∀ (e e_1 : Lean.Expr),
(Lean.Meta.Grind.EMatchTheoremConstraint.guard e = Lean.Meta.Grind.EMatchTheoremConstraint.guard e_1) = (e = e_1) | null | true |
CategoryTheory.eHom_whisker_cancel_inv | Mathlib.CategoryTheory.Enriched.Ordinary.Basic | ∀ (V : Type u') [inst : CategoryTheory.Category.{v', u'} V] [inst_1 : CategoryTheory.MonoidalCategory V] {C : Type u}
[inst_2 : CategoryTheory.Category.{v, u} C] [inst_3 : CategoryTheory.EnrichedOrdinaryCategory V C] {X Y Y₁ Z : C}
(α : Y ≅ Y₁),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCat... | null | true |
_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt.0.CFC.conjSqrt_ringInverse_conjSqrt._proof_1_3 | Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | ∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A]
[inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [inst_6 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
[inst_7 : NonnegSpectrumClass ℝ A] [inst_8 : SeparatelyContinuousMul A] [IsSemitopologicalRing ... | null | false |
_private.Init.Data.UInt.Bitwise.0.UInt64.zero_shiftLeft._simp_1_1 | Init.Data.UInt.Bitwise | ∀ {a b : UInt64}, (a = b) = (a.toBitVec = b.toBitVec) | null | false |
_private.Mathlib.Data.Fin.Tuple.Reflection.0.FinVec.Forall.match_1.eq_1 | Mathlib.Data.Fin.Tuple.Reflection | ∀ {α : Type u_2} (motive : (x : ℕ) → ((Fin x → α) → Prop) → Sort u_1) (P : (Fin 0 → α) → Prop)
(h_1 : (P : (Fin 0 → α) → Prop) → motive 0 P) (h_2 : (n : ℕ) → (P : (Fin (n + 1) → α) → Prop) → motive n.succ P),
(match 0, P with
| 0, P => h_1 P
| n.succ, P => h_2 n P) =
h_1 P | null | true |
CategoryTheory.Abelian.SpectralObject.zero₁_assoc | Mathlib.Algebra.Homology.SpectralObject.Basic | ∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{u_3, u_1} C]
[inst_1 : CategoryTheory.Category.{u_4, u_2} ι] [inst_2 : CategoryTheory.Abelian C]
(X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k)
(h : CategoryTheory.CategoryStruct.comp f g = fg) (n₀ n... | null | true |
Lean.Omega.IntList.gcd_eq_zero._simp_1 | Init.Omega.IntList | ∀ {xs : Lean.Omega.IntList}, (xs.gcd = 0) = ∀ x ∈ xs, x = 0 | null | false |
Module.Basis.range_extend | Mathlib.LinearAlgebra.Basis.VectorSpace | ∀ {K : Type u_3} {V : Type u_4} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {s : Set V}
(hs : LinearIndepOn K id s), Set.range ⇑(Module.Basis.extend hs) = hs.extend ⋯ | null | true |
Ring.DirectLimit.lift_of | Mathlib.Algebra.Colimit.Ring | ∀ {ι : Type u_1} [inst : Preorder ι] {G : ι → Type u_2} [inst_1 : (i : ι) → CommRing (G i)]
{f : (i j : ι) → i ≤ j → G i → G j} (P : Type u_3) [inst_2 : CommRing P] (g : (i : ι) → G i →+* P)
(Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x) (i : ι) (x : G i),
(Ring.DirectLimit.lift G f P g... | null | true |
NonUnitalSubsemiring.prodEquiv | Mathlib.RingTheory.NonUnitalSubsemiring.Basic | {R : Type u} →
{S : Type v} →
[inst : NonUnitalNonAssocSemiring R] →
[inst_1 : NonUnitalNonAssocSemiring S] →
(s : NonUnitalSubsemiring R) → (t : NonUnitalSubsemiring S) → ↥(s.prod t) ≃+* ↥s × ↥t | Product of non-unital subsemirings is isomorphic to their product as semigroups. | true |
_private.Lean.Meta.InferType.0.Lean.Meta.ArrowPropResult.casesOn | Lean.Meta.InferType | {motive : Lean.Meta.ArrowPropResult✝ → Sort u} →
(t : Lean.Meta.ArrowPropResult✝) →
motive Lean.Meta.ArrowPropResult.false✝ →
motive Lean.Meta.ArrowPropResult.true✝ →
motive Lean.Meta.ArrowPropResult.undef✝ → ((idx : ℕ) → motive (Lean.Meta.ArrowPropResult.bvar✝ idx)) → motive t | null | false |
Std.TreeMap.Raw.contains_iff_mem._simp_1 | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t : Std.TreeMap.Raw α β cmp} {k : α},
(t.contains k = true) = (k ∈ t) | null | false |
Lean.Elab.Command.InductiveElabStep2 | Lean.Elab.MutualInductive | Type | An intermediate step for mutual inductive elaboration. See `InductiveElabDescr`. | true |
_private.Mathlib.Tactic.CategoryTheory.Elementwise.0.Mathlib.Tactic.Elementwise.mkUnusedName.loop | Mathlib.Tactic.CategoryTheory.Elementwise | List Lean.Name → Lean.Name → optParam ℕ 0 → Lean.Name | null | true |
GradedAlgHom.mk.injEq | Mathlib.RingTheory.GradedAlgebra.AlgHom | ∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} {ι : Type u_4} [inst : CommSemiring R] [inst_1 : Semiring A]
[inst_2 : Semiring B] [inst_3 : Algebra R A] [inst_4 : Algebra R B] [inst_5 : DecidableEq ι] [inst_6 : AddMonoid ι]
{𝒜 : ι → Submodule R A} {ℬ : ι → Submodule R B} [inst_7 : GradedAlgebra 𝒜] [inst_8 : Grade... | null | true |
CategoryTheory.Monad.monadicOfHasPreservesReflexiveCoequalizersOfReflectsIsomorphisms._proof_1 | Mathlib.CategoryTheory.Monad.Monadicity | ∀ {C : Type u_3} {D : Type u_2} [inst : CategoryTheory.Category.{u_1, u_3} C]
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] {G : CategoryTheory.Functor D C} {F : CategoryTheory.Functor C D}
(adj : F ⊣ G) [CategoryTheory.Limits.HasReflexiveCoequalizers D] [G.ReflectsIsomorphisms]
[CategoryTheory.Monad.PreservesC... | null | false |
Lean.Parser.Module.prelude | Lean.Parser.Module.Syntax | Lean.Parser.Parser | null | true |
sigmaFinsuppEquivDFinsupp_symm_apply | Mathlib.Data.Finsupp.ToDFinsupp | ∀ {ι : Type u_1} {η : ι → Type u_4} {N : Type u_5} [inst : Zero N] (f : Π₀ (i : ι), η i →₀ N) (s : (i : ι) × η i),
(sigmaFinsuppEquivDFinsupp.symm f) s = (f s.fst) s.snd | null | true |
_private.Batteries.Data.Array.Lemmas.0.Array.extract_append_of_stop_le_size_left._proof_1_14 | Batteries.Data.Array.Lemmas | ∀ {α : Type u_1} {j i : ℕ} {a b : Array α}, j ≤ a.size → (a ++ b).extract i j = a.extract i j | null | false |
AlgebraicGeometry.specOrderIsoPrimeSpectrum_apply | Mathlib.AlgebraicGeometry.Scheme | ∀ (R : CommRingCat) (x : ↥(AlgebraicGeometry.Spec R)),
(AlgebraicGeometry.specOrderIsoPrimeSpectrum R) x = OrderDual.toDual x | null | true |
_private.Mathlib.Analysis.InnerProductSpace.Symmetric.0.Submodule.isSymmetric_projection_iff._simp_1_1 | Mathlib.Analysis.InnerProductSpace.Symmetric | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{x y : E}, (inner 𝕜 x y = 0) = (inner 𝕜 y x = 0) | null | false |
UInt8.ofBitVec_shiftRight_mod | Init.Data.UInt.Bitwise | ∀ (a : BitVec 8) (b : ℕ), { toBitVec := a >>> (b % 8) } = { toBitVec := a } >>> UInt8.ofNat b | null | true |
List.swapTR.go._f | Batteries.Data.List.Basic | {α : Type u_1} →
List α → (a : List α) → List.below (motive := fun a => ℕ → ℕ → Array α → List α) a → ℕ → ℕ → Array α → List α | null | false |
Std.Time.TimeZone.convertTZif | Std.Time.Zoned.Database.Basic | Std.Time.TimeZone.TZif.TZif → String → Except String Std.Time.TimeZone.ZoneRules | Converts a `TZif.TZif` structure to a `ZoneRules` structure.
| true |
_private.Init.Data.Iterators.Lemmas.Consumers.Collect.0.Std.Iter.atIdxSlow?.match_1.eq_2 | Init.Data.Iterators.Lemmas.Consumers.Collect | ∀ {α β : Type u_1} [inst : Std.Iterator α Id β] (it : Std.Iter β)
(motive :
(n : ℕ) →
((a' : Std.Iter β) →
(b' : ℕ) →
InvImage (Prod.Lex WellFoundedRelation.rel Std.IterM.TerminationMeasures.Productive.Rel)
(fun p => (p.snd, p.fst.finitelyManySkips!)) ⟨a', b'⟩ ⟨it, n⟩ →... | null | true |
Array.getElem_toList | Init.Data.Array.Basic | ∀ {α : Type u} {xs : Array α} {i : ℕ} (h : i < xs.toList.length), xs.toList[i] = xs[i] | null | true |
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