name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
Equiv.Perm.instDecidableRelSameCycleInv | Mathlib.GroupTheory.Perm.Cycle.Basic | {α : Type u_2} → (f : Equiv.Perm α) → [DecidableRel f.SameCycle] → DecidableRel f⁻¹.SameCycle | null | true |
LocallyPathConnectedSpace.mk._flat_ctor | Mathlib.Topology.Connected.LocallyPathConnected | ∀ {X : Type u_4} [inst : TopologicalSpace X],
(∀ (x : X), (nhds x).HasBasis (fun s => s ∈ nhds x ∧ IsPathConnected s) id) → LocallyPathConnectedSpace X | null | false |
MonomialOrder.degree_prod_of_regular | Mathlib.RingTheory.MvPolynomial.MonomialOrder | ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommSemiring R] {ι : Type u_3} {P : ι → MvPolynomial σ R}
{s : Finset ι}, (∀ i ∈ s, IsRegular (m.leadingCoeff (P i))) → m.degree (∏ i ∈ s, P i) = ∑ i ∈ s, m.degree (P i) | null | true |
Lean.PersistentHashSet.recOn | Lean.Data.PersistentHashSet | {α : Type u} →
[inst : BEq α] →
[inst_1 : Hashable α] →
{motive : Lean.PersistentHashSet α → Sort u_1} →
(t : Lean.PersistentHashSet α) → ((set : Lean.PersistentHashMap α Unit) → motive { set := set }) → motive t | null | false |
Cardinal.mk_embedding_le_arrow | Mathlib.SetTheory.Cardinal.Arithmetic | ∀ (α : Type u) (β' : Type v), Cardinal.mk (α ↪ β') ≤ Cardinal.mk (α → β') | null | true |
CategoryTheory.Subobject.lower | Mathlib.CategoryTheory.Subobject.Basic | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{X : C} →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
{Y : D} →
CategoryTheory.Functor (CategoryTheory.MonoOver X) (CategoryTheory.MonoOver Y) →
CategoryTheory.Functor (CategoryTheor... | Any functor `MonoOver X ⥤ MonoOver Y` descends to a functor
`Subobject X ⥤ Subobject Y`, because `MonoOver Y` is thin. | true |
_private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.realizeExtIffTheorem | Lean.Elab.Tactic.Ext | Lean.Name → Lean.Elab.TermElabM Lean.Name | Given an 'ext' theorem, ensures that there is an iff version of the theorem (if possible),
without validating any pre-existing theorems.
Returns the name of the 'ext_iff' theorem.
| true |
Matrix.PosDef.of_toQuadraticForm' | Mathlib.LinearAlgebra.Matrix.PosDef | ∀ {n : Type u_2} [inst : Fintype n] {R : Type u_5} [inst_1 : CommRing R] [inst_2 : PartialOrder R] [inst_3 : StarRing R]
[TrivialStar R] [inst_5 : DecidableEq n] {M : Matrix n n R},
M.IsSymm → QuadraticMap.PosDef M.toQuadraticForm' → M.PosDef | null | true |
_private.Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer.0.Equiv.Perm.count_le_one_of_centralizer_le_alternating._simp_1_6 | Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | ∀ {α : Type u_1} {β : Type v} {f : α → β} {b : β} {s : Multiset α}, (b ∈ Multiset.map f s) = ∃ a ∈ s, f a = b | null | false |
SemigroupIdeal.subset_closure | Mathlib.Algebra.Group.Ideal | ∀ {M : Type u_1} [inst : Mul M] {s : Set M}, s ⊆ ↑(SemigroupIdeal.closure s) | null | true |
Lean.Expr.FoldRelevantConstantsImpl.instInhabitedArgKind.default | Lean.LibrarySuggestions.Basic | Lean.Expr.FoldRelevantConstantsImpl.ArgKind✝ | null | true |
Lean.Grind.NoopConfig.funCC._default | Init.Grind.Config | Bool | null | false |
ExteriorAlgebra.instGradedMonoidNatSubmoduleExteriorPower | Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | ∀ (R : Type u_1) (M : Type u_2) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
SetLike.GradedMonoid fun i => ⋀[R]^i M | null | true |
AlgebraicGeometry.pointsPi.eq_1 | Mathlib.AlgebraicGeometry.PointsPi | ∀ {ι : Type u} (R : ι → CommRingCat) (X : AlgebraicGeometry.Scheme) (f : AlgebraicGeometry.Spec ↧((i : ι) → ↑(R i)) ⟶ X)
(i : ι),
AlgebraicGeometry.pointsPi R X f i =
CategoryTheory.CategoryStruct.comp
(AlgebraicGeometry.Spec.map (CommRingCat.ofHom (Pi.evalRingHom (fun x => ↑(R x)) i))) f | null | true |
Filter.HasBasis.tendstoUniformlyOn_iff_of_uniformity | Mathlib.Topology.UniformSpace.UniformConvergence | ∀ {α : Type u_1} {β : Type u_2} [inst : UniformSpace β] {X : Type u_5} {ιβ : Type u_8} {F : X → α → β} {f : α → β}
{l : Filter X} {s : Set α} {pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)},
(uniformity β).HasBasis pβ sβ →
(TendstoUniformlyOn F f l s ↔ ∀ (i : ιβ), pβ i → ∀ᶠ (n : X) in l, ∀ x ∈ s, (f x, F n x) ∈ sβ i) | An analogue of `Filter.HasBasis.tendsto_right_iff` for `TendstoUniformlyOn`. | true |
Algebra.TensorProduct.lTensor_ker | Mathlib.LinearAlgebra.TensorProduct.RightExactness | ∀ {R : Type u_4} [inst : CommRing R] {A : Type u_6} {C : Type u_8} {D : Type u_9} [inst_1 : Ring A] [inst_2 : Ring C]
[inst_3 : Ring D] [inst_4 : Algebra R A] [inst_5 : Algebra R C] [inst_6 : Algebra R D] (g : C →ₐ[R] D),
Function.Surjective ⇑g →
RingHom.ker (Algebra.TensorProduct.map (AlgHom.id R A) g) =
... | If `g` is surjective, then the kernel of `(id A) ⊗ g` is generated by the kernel of `g` | true |
Convex.add_smul_sub_mem_interior' | Mathlib.Analysis.Convex.Topology | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Field 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommGroup E]
[inst_3 : Module 𝕜 E] [inst_4 : TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
[AddRightMono 𝕜] {s : Set E},
Convex 𝕜 s → ∀ {x y : E}, x ∈ closure s → y ∈ interior s → ∀ {t : 𝕜}, t... | If `x ∈ closure s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`.
| true |
_private.Lean.Elab.MutualInductive.0.Lean.Elab.Command.inferResultingUniverse.match_4 | Lean.Elab.MutualInductive | (motive : Lean.Level × ℤ → Sort u_1) →
(x : Lean.Level × ℤ) → ((level : Lean.Level) → (k : ℤ) → motive (level, k)) → motive x | null | false |
Std.DTreeMap.Raw.Const.getKeyD_insertMany_list_of_mem | Std.Data.DTreeMap.Raw.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : Std.DTreeMap.Raw α (fun x => β) cmp} [Std.TransCmp cmp],
t.WF →
∀ {l : List (α × β)} {k k' fallback : α},
cmp k k' = Ordering.eq →
List.Pairwise (fun a b => ¬cmp a.1 b.1 = Ordering.eq) l →
k ∈ List.map Prod.fst l → (Std.DTreeMap.Raw... | null | true |
Lean.Meta.Grind.SolverExtension.mbtc | Lean.Meta.Tactic.Grind.Types | {σ : Type} → Lean.Meta.Grind.SolverExtension σ → Lean.Meta.Grind.GoalM Bool | null | true |
isPrimitiveRoot_of_mem_primitiveRoots | Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | ∀ {R : Type u_4} {k : ℕ} [inst : CommRing R] [inst_1 : IsDomain R] {ζ : R}, ζ ∈ primitiveRoots k R → IsPrimitiveRoot ζ k | null | true |
CategoryTheory.Subfunctor.preimage._proof_2 | Mathlib.CategoryTheory.Subfunctor.Image | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {F F' : CategoryTheory.Functor C (Type u_3)}
(G : CategoryTheory.Subfunctor F) (p : F' ⟶ F) {U V : C} (f : U ⟶ V),
⇑(CategoryTheory.ConcreteCategory.hom (p.app U)) ⁻¹' G.obj U ⊆
⇑(CategoryTheory.ConcreteCategory.hom (F'.map f)) ⁻¹' ⇑(CategoryTheory.... | null | false |
LieHom.mem_range | Mathlib.Algebra.Lie.Subalgebra | ∀ {R : Type u} {L : Type v} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] {L₂ : Type w}
[inst_3 : LieRing L₂] [inst_4 : LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L₂), x ∈ f.range ↔ ∃ y, f y = x | null | true |
quotAdjoinEquivQuotMap.congr_simp | Mathlib.NumberTheory.KummerDedekind | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S} {I : Ideal R}
(hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)),
quotAdjoinEquivQuotMap hx h_alg = quotAdjoinEquivQuotMap hx h_alg | null | true |
MeasureTheory.Filtration.HasLimitProcess.comp₂ | Mathlib.Probability.Process.LimitProcess | ∀ {ι : Type u_1} {Ω : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} [inst : Preorder ι]
{mΩ : MeasurableSpace Ω} [inst_1 : TopologicalSpace E] [inst_2 : TopologicalSpace F] [inst_3 : TopologicalSpace G]
{P : MeasureTheory.Measure Ω} {𝓕 : MeasureTheory.Filtration ι mΩ} {X : ι → Ω → E} {Z : ι → Ω → F} {f : ... | null | true |
IsLocalization.smul_toInvSubmonoid | Mathlib.RingTheory.Localization.InvSubmonoid | ∀ {R : Type u_1} [inst : CommRing R] (M : Submonoid R) (S : Type u_2) [inst_1 : CommRing S] [inst_2 : Algebra R S]
[inst_3 : IsLocalization M S] (m : ↥M), m • ↑((IsLocalization.toInvSubmonoid M S) m) = 1 | null | true |
Rat.padicValuation.congr_simp | Mathlib.NumberTheory.Padics.PadicNumbers | ∀ (p p_1 : ℕ) (e_p : p = p_1) [inst : Fact (Nat.Prime p)], Rat.padicValuation p = Rat.padicValuation p_1 | null | true |
_private.Mathlib.RingTheory.DiscreteValuationRing.TFAE.0.IsDedekindDomain.isPrincipalIdealRing._proof_2 | Mathlib.RingTheory.DiscreteValuationRing.TFAE | 1 ≠ 0 | null | false |
Lean.Meta.Origin.other.noConfusion | Lean.Meta.Tactic.Simp.SimpTheorems | {P : Sort u} →
{name name' : Lean.Name} → Lean.Meta.Origin.other name = Lean.Meta.Origin.other name' → (name = name' → P) → P | null | false |
Lean.Elab.InfoTree.brecOn_4 | Lean.Elab.InfoTree.Types | {motive_1 : Lean.Elab.InfoTree → Sort u} →
{motive_2 : Lean.PersistentArray Lean.Elab.InfoTree → Sort u} →
{motive_3 : Lean.PersistentArrayNode Lean.Elab.InfoTree → Sort u} →
{motive_4 : Array Lean.Elab.InfoTree → Sort u} →
{motive_5 : Array (Lean.PersistentArrayNode Lean.Elab.InfoTree) → Sort u} →
... | null | false |
Turing.TM2to1.addBottom_modifyNth | Mathlib.Computability.TuringMachine.StackTuringMachine | ∀ {K : Type u_1} {Γ : K → Type u_2} (f : ((k : K) → Option (Γ k)) → (k : K) → Option (Γ k))
(L : Turing.ListBlank ((k : K) → Option (Γ k))) (n : ℕ),
Turing.ListBlank.modifyNth (fun a => (a.1, f a.2)) n (Turing.TM2to1.addBottom L) =
Turing.TM2to1.addBottom (Turing.ListBlank.modifyNth f n L) | null | true |
_private.Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality.0.groupCohomology.cochainsMap_f_map_mono._simp_1_1 | Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | ∀ {R : Type u} [inst : Ring R] {X Y : ModuleCat R} (f : X ⟶ Y),
CategoryTheory.Mono f = Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom f) | null | false |
ContinuousOn.strictMonoOn_of_injOn_Icc' | Mathlib.Topology.Order.IntermediateValue | ∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α]
[DenselyOrdered α] {δ : Type u_1} [inst_4 : LinearOrder δ] [inst_5 : TopologicalSpace δ] [OrderClosedTopology δ]
{a b : α} {f : α → δ},
a ≤ b →
ContinuousOn f (Set.Icc a b) →
Set.InjOn f (Set.Icc a... | Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone
or antitone (increasing or decreasing). | true |
CategoryTheory.Limits.cokernelOrderHom._proof_2 | Mathlib.CategoryTheory.Subobject.Limits | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_2 : CategoryTheory.Limits.HasCokernels C] (X : C) (P Q : CategoryTheory.Subobject X),
P ≤ Q →
CategoryTheory.Subobject.lift (fun x f x_1 => CategoryTheory.Subobject.mk (CategoryTheory.Limits.... | null | false |
_private.Mathlib.Data.Fin.Tuple.Reflection.0.FinVec.sum.match_1.eq_1 | Mathlib.Data.Fin.Tuple.Reflection | ∀ {α : Type u_2} (motive : (x : ℕ) → (Fin x → α) → Sort u_1) (x : Fin 0 → α) (h_1 : (x : Fin 0 → α) → motive 0 x)
(h_2 : (v : Fin 1 → α) → motive 1 v) (h_3 : (n : ℕ) → (v : Fin (n + 2) → α) → motive n.succ.succ v),
(match 0, x with
| 0, x => h_1 x
| 1, v => h_2 v
| n.succ.succ, v => h_3 n v) =
h_1 x | null | true |
CategoryTheory.IsFiltered.instSmallCategorySmallFilteredIntermediate._aux_1 | Mathlib.CategoryTheory.Filtered.Small | {C : Type u_4} →
[inst : CategoryTheory.Category.{u_2, u_4} C] →
[inst_1 : CategoryTheory.IsFilteredOrEmpty C] →
{D : Type u_1} →
[inst_2 : CategoryTheory.Category.{u_3, u_1} D] →
(F : CategoryTheory.Functor D C) → Quiver (CategoryTheory.IsFiltered.SmallFilteredIntermediate F) | null | false |
ConvexCone.coe_closure._simp_1 | Mathlib.Analysis.Convex.Cone.Closure | ∀ {𝕜 : Type u_1} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] {E : Type u_2} [inst_2 : AddCommMonoid E]
[inst_3 : TopologicalSpace E] [inst_4 : ContinuousAdd E] [inst_5 : SMul 𝕜 E] [inst_6 : ContinuousConstSMul 𝕜 E]
(K : ConvexCone 𝕜 E), closure ↑K = ↑K.closure | null | false |
Cardinal.range_aleph | Mathlib.SetTheory.Cardinal.Aleph | Set.range ⇑Cardinal.aleph = Set.Ici Cardinal.aleph0 | null | true |
_private.Mathlib.Order.ConditionallyCompleteLattice.Finset.0.Set.Finite.ciSup_lt_iff._simp_1_1 | Mathlib.Order.ConditionallyCompleteLattice.Finset | ∀ {α : Type u} {ι : Sort u_1} {f : ι → α} {x : α}, (x ∈ Set.range f) = ∃ y, f y = x | null | false |
Turing.ToPartrec.Cfg.halt.noConfusion | Mathlib.Computability.TuringMachine.Config | {P : Sort u} → {a a' : List ℕ} → Turing.ToPartrec.Cfg.halt a = Turing.ToPartrec.Cfg.halt a' → (a = a' → P) → P | null | false |
lt_emultiplicity_of_lt_multiplicity | Mathlib.RingTheory.Multiplicity | ∀ {α : Type u_1} [inst : Monoid α] {a b : α} {n : ℕ}, n < multiplicity a b → ↑n < emultiplicity a b | null | true |
Std.Http.Internal.ChunkedBuffer.mk.sizeOf_spec | Std.Http.Internal.ChunkedBuffer | ∀ (data : Array ByteArray) (size : ℕ), sizeOf { data := data, size := size } = 1 + sizeOf data + sizeOf size | null | true |
Num.pred_succ | Mathlib.Data.Num.ZNum | ∀ (n : ZNum), n.pred.succ = n | null | true |
BitVec.sshiftRight_eq_sshiftRightRec | Init.Data.BitVec.Bitblast | ∀ {w₁ w₂ i : ℕ} (x : BitVec w₁) (y : BitVec w₂), (x.sshiftRight' y).getLsbD i = (x.sshiftRightRec y (w₂ - 1)).getLsbD i | Show that `x.sshiftRight y` can be written in terms of `sshiftRightRec`.
This can be unfolded in terms of `sshiftRightRec_zero_eq`, `sshiftRightRec_succ_eq` for bit blasting.
| true |
Graph.ext | Mathlib.Combinatorics.Graph.Basic | ∀ {α : Type u_1} {β : Type u_2} {G₁ G₂ : Graph α β},
G₁.vertexSet = G₂.vertexSet → (∀ (e : β) (x y : α), G₁.IsLink e x y ↔ G₂.IsLink e x y) → G₁ = G₂ | Two graphs with the same vertex set and binary incidences are equal.
(We use this as the default extensionality lemma rather than adding `@[ext]`
to the definition of `Graph`, so it doesn't require equality of the edge sets.) | true |
IsometryEquiv.instGroup | Mathlib.Topology.MetricSpace.Isometry | {α : Type u} → [inst : PseudoEMetricSpace α] → Group (α ≃ᵢ α) | The group of isometries. | true |
and_self | Init.SimpLemmas | ∀ (p : Prop), (p ∧ p) = p | null | true |
Std.Tactic.BVDecide.BVExpr.bitblast.instLawfulVecOperatorArbitraryShiftTargetBlastArithShiftRight | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Operations.ShiftRight | ∀ {α : Type} [inst : Hashable α] [inst_1 : DecidableEq α],
Std.Sat.AIG.LawfulVecOperator α Std.Sat.AIG.ArbitraryShiftTarget fun {len} =>
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight | null | true |
classifyingSpaceUniversalCover.compForgetAugmented | Mathlib.RepresentationTheory.Homological.Resolution | (G : Type u) → [Monoid G] → CategoryTheory.SimplicialObject.Augmented (Type u) | The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `Fin 1 → G` to the terminal object in `Type u`. | true |
isIntegralCurveOn_univ | Mathlib.Analysis.ODE.Basic | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {γ : ℝ → E} {v : ℝ → E → E},
IsIntegralCurveOn γ v Set.univ ↔ IsIntegralCurve γ v | null | true |
ISize.xor_eq_zero_iff._simp_1 | Init.Data.SInt.Bitwise | ∀ {a b : ISize}, (a ^^^ b = 0) = (a = b) | null | false |
Lean.Meta.matchFalse | Lean.Meta.MatchUtil | Lean.Expr → Lean.MetaM Bool | null | true |
_private.Mathlib.Algebra.Category.ModuleCat.ChangeOfRings.0.ModuleCat.extendScalars_assoc'._simp_1_1 | Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D)
{G H K : CategoryTheory.Functor D E} (α : G ⟶ H) (β : H ⟶ K) {Z : CategoryTheory.Functor C E} (h : F.co... | null | false |
_private.Init.Data.List.MinMaxOn.0.List.minOn.eq_1 | Init.Data.List.MinMaxOn | ∀ {β : Type u_1} {α : Type u_2} [inst : LE β] [inst_1 : DecidableLE β] (f : α → β) (x : α) (xs : List α)
(h_2 : x :: xs ≠ []), List.minOn f (x :: xs) h_2 = List.foldl (minOn f) x xs | null | true |
_private.Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization.0.Polynomial.natDegree_of_dvd_cyclotomic_of_irreducible_of_monic._simp_1_1 | Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | ∀ {R : Type u_1} [inst : CommRing R] {n : ℕ} [IsDomain R] [NeZero ↑n] {μ : R},
IsPrimitiveRoot μ n = (Polynomial.cyclotomic n R).IsRoot μ | null | false |
LinearMap.toLinearIsometry | Mathlib.Analysis.Normed.Operator.LinearIsometry | {R : Type u_1} →
{R₂ : Type u_2} →
{E : Type u_4} →
{E₂ : Type u_5} →
[inst : Semiring R] →
[inst_1 : Semiring R₂] →
{σ₁₂ : R →+* R₂} →
[inst_2 : SeminormedAddCommGroup E] →
[inst_3 : SeminormedAddCommGroup E₂] →
[inst_4 : Module ... | Construct a `LinearIsometry` from a `LinearMap` satisfying `Isometry`. | true |
Nat.Internal.Linear.ExprCnstr.noConfusionType | Init.Data.Nat.Internal.Linear | Sort u → Nat.Internal.Linear.ExprCnstr → Nat.Internal.Linear.ExprCnstr → Sort u | null | false |
Finset.isCountablyCompact_biUnion | Mathlib.Topology.Compactness.CountablyCompact | ∀ {ι : Type u_1} {E : Type u_2} [inst : TopologicalSpace E] (s : Finset ι) {f : ι → Set E},
(∀ i ∈ s, IsCountablyCompact (f i)) → IsCountablyCompact (⋃ i ∈ s, f i) | A finite union of countably compact sets is countably compact. | true |
Submonoid.distribMulAction | Mathlib.Algebra.Group.Submonoid.DistribMulAction | {M : Type u_1} →
{α : Type u_2} →
[inst : Monoid M] → [inst_1 : AddMonoid α] → [DistribMulAction M α] → (S : Submonoid M) → DistribMulAction (↥S) α | The action by a submonoid is the action by the underlying monoid. | true |
Int.divisorsAntidiagonal_one | Mathlib.NumberTheory.Divisors | Int.divisorsAntidiag 1 = {(1, 1), (-1, -1)} | null | true |
CoeDep.noConfusion | Init.Coe | {P : Sort u_1} →
{α : Sort u} →
{x : α} →
{β : Sort v} →
{t : CoeDep α x β} →
{α' : Sort u} →
{x' : α'} →
{β' : Sort v} → {t' : CoeDep α' x' β'} → α = α' → x ≍ x' → β = β' → t ≍ t' → CoeDep.noConfusionType P t t' | null | false |
Lex.instSemifield._proof_2 | Mathlib.Algebra.Field.Basic | ∀ {K : Type u_1} [inst : Semifield K], autoParam (∀ (a : Lex K), a ^ 0 = 1) DivInvMonoid.zpow_zero'._autoParam | null | false |
Algebra.PreSubmersivePresentation.ofAlgEquiv | Mathlib.RingTheory.Extension.Presentation.Submersive | {R : Type u} →
{S : Type v} →
{ι : Type w} →
{σ : Type t} →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
Algebra.PreSubmersivePresentation R S ι σ →
{T : Type u_1} →
[inst_3 : CommRing T] →
... | Transport a pre-submersive presentation along an algebra isomorphism. | true |
AffineSubspace.coe_vsub._simp_1 | Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | ∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
[inst_3 : AddTorsor V P] (s : AffineSubspace k P) [inst_4 : Nonempty ↥s] (a b : ↥s), ↑a -ᵥ ↑b = ↑(a -ᵥ b) | null | false |
AkraBazziRecurrence.eventually_one_add_smoothingFn_pos | Mathlib.Computability.AkraBazzi.SumTransform | ∀ᶠ (n : ℕ) in Filter.atTop, 0 < 1 + AkraBazziRecurrence.smoothingFn ↑n | null | true |
wrapSimpDischarger | Mathlib.Util.DischargerAsTactic | Lean.Meta.Simp.Discharge → Lean.Elab.Tactic.TacticM Unit | Wrap a simp discharger (a function `Expr → SimpM (Option Expr)`) as a tactic,
so that it can be passed as an argument to `simp (discharger := foo)`.
This is inverse to `mkDischargeWrapper`. | true |
_private.Mathlib.Tactic.FunProp.Core.0.Mathlib.Meta.FunProp.tryTheoremWithHint?.match_1 | Mathlib.Tactic.FunProp.Core | (motive : ℕ × Lean.Expr → Sort u_1) → (x : ℕ × Lean.Expr) → ((id : ℕ) → (v : Lean.Expr) → motive (id, v)) → motive x | null | false |
instLieGroupOfNatWithTopENatOfIsTopologicalGroup | Mathlib.Geometry.Manifold.Algebra.LieGroup | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
[inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4}
[inst_4 : TopologicalSpace G] [inst_5 : ChartedSpace H G] [inst_6 : Group G] [IsTopologicalGroup ... | null | true |
Lean.Grind.Order.eq_of_le_of_le_0 | Init.Grind.Order | ∀ {α : Type u_1} [inst : LE α] [Std.IsPartialOrder α] [inst_2 : Lean.Grind.Ring α] {a b : α},
a ≤ b + ↑0 → b ≤ a + ↑0 → a = b | null | true |
UpperHalfPlane.IsZeroAtImInfty.eq_1 | Mathlib.NumberTheory.ModularForms.BoundedAtCusp | ∀ {α : Type u_1} [inst : Zero α] [inst_1 : TopologicalSpace α] (f : UpperHalfPlane → α),
UpperHalfPlane.IsZeroAtImInfty f = UpperHalfPlane.atImInfty.ZeroAtFilter f | null | true |
IsMulTorsionFree.eq_of_pow_eq_pow_of_commute | Mathlib.Algebra.Group.Monoid | ∀ {M : Type u_2} {inst : Monoid M} [self : IsMulTorsionFree M] ⦃n : ℕ⦄,
n ≠ 0 → ∀ ⦃a b : M⦄, Commute a b → a ^ n = b ^ n → a = b | null | true |
Irreducible.maximalIdeal_pow_eq_setOf_le_v_coe_pow | Mathlib.RingTheory.DiscreteValuationRing.Basic | ∀ {K : Type u_1} {Γ₀ : Type u_2} [inst : Field K] [inst_1 : LinearOrderedCommGroupWithZero Γ₀] (v : Valuation K Γ₀)
[inst_2 : IsDiscreteValuationRing ↥v.integer] {ϖ : ↥v.integer},
Irreducible ϖ → ∀ (n : ℕ), ↑(IsLocalRing.maximalIdeal ↥v.integer ^ n) = {y | v ↑y ≤ v ↑ϖ ^ n} | **Alias** of `Irreducible.maximalIdeal_pow_eq_setOfPred_le_v_coe_pow`. | true |
Homeomorph.prodComm_trans_prodComm | Mathlib.Topology.Constructions.SumProd | ∀ (X : Type u) (Y : Type v) [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y],
(Homeomorph.prodComm X Y).trans (Homeomorph.prodComm Y X) = Homeomorph.refl (X × Y) | null | true |
_private.Mathlib.RingTheory.FractionalIdeal.Operations.0.FractionalIdeal.eq_zero_or_one._simp_1_1 | Mathlib.RingTheory.FractionalIdeal.Operations | ∀ {R : Type u_1} [inst : CommRing R] (S : Submonoid R) {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
{x : P}, (x ∈ 1) = ∃ x', (algebraMap R P) x' = x | null | false |
Lean.Meta.ExtractLetsConfig.merge | Init.MetaTypes | Lean.Meta.ExtractLetsConfig → Bool | If true (default: true), reuse local declarations that have syntactically equal values.
Note that even when false, the caching strategy for `extract_lets` may result in fewer extracted let bindings than expected. | true |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.isNone_minEntry?_eq_isEmpty._simp_1_3 | Std.Data.Internal.List.Associative | ∀ {α : Type u_1} {l : List α}, (l.isEmpty = true) = (l = []) | null | false |
SeparationQuotient.continuousOn_lift₂._simp_1 | Mathlib.Topology.Inseparable | ∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
[inst_2 : TopologicalSpace Z] {f : X → Y → Z}
{hf : ∀ (a : X) (b : Y) (c : X) (d : Y), Inseparable a c → Inseparable b d → f a b = f c d}
{s : Set (SeparationQuotient X × SeparationQuotient Y)},
ContinuousOn... | null | false |
_private.Mathlib.LinearAlgebra.Matrix.Notation.0.Matrix.diagonal_fin_three._simp_1_2 | Mathlib.LinearAlgebra.Matrix.Notation | ∀ {n : ℕ} {P : Fin (n + 1) → Prop}, (∀ (i : Fin (n + 1)), P i) = (P 0 ∧ ∀ (i : Fin n), P i.succ) | null | false |
ContinuousMultilinearMap.smulRightL._proof_8 | Mathlib.Analysis.Normed.Module.Multilinear.Basic | ∀ (𝕜 : Type u_1) (G : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup G]
[inst_2 : NormedSpace 𝕜 G], ContinuousSMul 𝕜 G | null | false |
Topology.RelCWComplex.finiteType_mkFiniteType | Mathlib.Topology.CWComplex.Classical.Finite | ∀ {X : Type u} [inst : TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : ℕ → Type u)
(map : (n : ℕ) → cell n → PartialEquiv (Fin n → ℝ) X) (finite_cell : ∀ (n : ℕ), Finite (cell n))
(source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = Metric.ball 0 1)
(continuousOn : ∀ (n : ℕ) (i : cell n), Contin... | A CW complex that was constructed using `RelCWComplex.mkFiniteType` is of finite type. | true |
TopHomClass.rec | Mathlib.Order.Hom.Bounded | {F : Type u_6} →
{α : Type u_7} →
{β : Type u_8} →
[inst : Top α] →
[inst_1 : Top β] →
[inst_2 : FunLike F α β] →
{motive : TopHomClass F α β → Sort u} →
((map_top : ∀ (f : F), f ⊤ = ⊤) → motive ⋯) → (t : TopHomClass F α β) → motive t | null | false |
List.getLast_range' | Init.Data.List.Nat.Range | ∀ {s n : ℕ} (h : List.range' s n ≠ []), (List.range' s n).getLast h = s + n - 1 | null | true |
AddLocalization.r'._proof_4 | Mathlib.GroupTheory.MonoidLocalization.Basic | ∀ {M : Type u_1} [inst : AddCommMonoid M] (S : AddSubmonoid M) {a b c : M × ↥S} (t₁ : ↥S),
↑t₁ + (↑b.2 + a.1) = ↑t₁ + (↑a.2 + b.1) →
∀ (t₂ : ↥S), ↑t₂ + ↑c.2 + (↑t₁ + (↑b.2 + a.1)) = ↑t₁ + ↑a.2 + (↑t₂ + (↑c.2 + b.1)) | null | false |
CochainComplex.mkHomAux | Mathlib.Algebra.Homology.HomologicalComplex | {V : Type u} →
[inst : CategoryTheory.Category.{v, u} V] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] →
(P Q : CochainComplex V ℕ) →
(zero : P.X 0 ⟶ Q.X 0) →
(one : P.X 1 ⟶ Q.X 1) →
CategoryTheory.CategoryStruct.comp zero (Q.d 0 1) = CategoryTheory.CategoryStruct.comp ... | An auxiliary construction for `mkHom`.
Here we build by induction a family of commutative squares,
but don't require at the type level that these successive commutative squares actually agree.
They do in fact agree, and we then capture that at the type level (i.e. by constructing a chain map)
in `mkHom`.
| true |
IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation' | Mathlib.RingTheory.DedekindDomain.AdicValuation | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDedekindDomain R] {K : Type u_2} [inst_2 : Field K]
[inst_3 : Algebra R K] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (k : K),
Valued.v
(IsDedekindDomain.HeightOneSpectrum.adicCompletion.ofCompletion
↑(WithVal.toVal (IsDede... | The valuation on the completion agrees with the global valuation on elements of the field. | true |
Equiv.toHomeomorphOfContinuousClosed_symm_apply | Mathlib.Topology.Homeomorph.Defs | ∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (e : X ≃ Y)
(h₁ : Continuous ⇑e) (h₂ : IsClosedMap ⇑e), ⇑(e.toHomeomorphOfContinuousClosed h₁ h₂).symm = ⇑e.symm | null | true |
Matroid.eRk_le_encard | Mathlib.Combinatorics.Matroid.Rank.ENat | ∀ {α : Type u_1} (M : Matroid α) (X : Set α), M.eRk X ≤ X.encard | null | true |
FintypeCat.botTopology | Mathlib.Topology.Category.Profinite.Basic | (A : FintypeCat) → TopologicalSpace A.obj | Finite types are given the discrete topology. | true |
Std.DTreeMap.Internal.Impl.toListModel_inner | Std.Data.DTreeMap.Internal.Def | ∀ {α : Type u} {β : α → Type v} {sz : ℕ} {k : α} {v : β k} {l r : Std.DTreeMap.Internal.Impl α β},
(Std.DTreeMap.Internal.Impl.inner sz k v l r).toListModel = l.toListModel ++ ⟨k, v⟩ :: r.toListModel | null | true |
instCompleteAtomicBooleanAlgebraLanguage | Mathlib.Computability.Language | (α : Type u_1) → CompleteAtomicBooleanAlgebra (Language α) | null | true |
Real.toNNReal_le_ofNat._simp_1 | Mathlib.Basic.NNReal.Defs | ∀ {r : ℝ} {n : ℕ} [inst : n.AtLeastTwo], (r.toNNReal ≤ OfNat.ofNat n) = (r ≤ ↑n) | null | false |
NonUnitalStarAlgebra.range_eq_top | Mathlib.Algebra.Star.NonUnitalSubalgebra | ∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
[inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : NonUnitalSemiring B] [inst_6 : StarRing B] [inst_7 : Module R B]
[inst_8 : IsScalarTower R B B] [inst_9 : SMulCommClass R B B] [inst_10 : StarMo... | null | true |
_private.Mathlib.Analysis.Analytic.Within.0.analyticWithinAt_of_singleton_mem._simp_1_7 | Mathlib.Analysis.Analytic.Within | ∀ {α : Type u} (x : α), (x ∈ Set.univ) = True | null | false |
IsSepClosed.of_exists_root | Mathlib.FieldTheory.IsSepClosed | ∀ (k : Type u) [inst : Field k],
(∀ (p : Polynomial k), p.Monic → Irreducible p → p.Separable → ∃ x, Polynomial.eval x p = 0) → IsSepClosed k | null | true |
Subtype.isEmpty_of_false | Mathlib.Basic.IsEmpty.Defs | ∀ {α : Sort u} {p : α → Prop}, (∀ (a : α), ¬p a) → IsEmpty (Subtype p) | Subtypes by an all-false predicate are empty. | true |
ContinuousLinearMap.range_coeFn_eq | Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | ∀ {R₁ : Type u_1} {R₂ : Type u_2} [inst : Semiring R₁] [inst_1 : Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4}
[inst_2 : TopologicalSpace M₁] [inst_3 : AddCommMonoid M₁] {M₂ : Type u_6} [inst_4 : TopologicalSpace M₂]
[inst_5 : AddCommMonoid M₂] [inst_6 : Module R₁ M₁] [inst_7 : Module R₂ M₂],
Set.range DFunLike.... | null | true |
Turing.PartrecToTM2.instDecidableEqCont'.decEq._proof_5 | Mathlib.Computability.TuringMachine.ToPartrec | ∀ (a : Turing.ToPartrec.Code) (a_1 : Turing.PartrecToTM2.Cont'),
¬Turing.PartrecToTM2.Cont'.halt = Turing.PartrecToTM2.Cont'.comp a a_1 | null | false |
Lean.CollectMVars.State.rec | Lean.Util.CollectMVars | {motive : Lean.CollectMVars.State → Sort u} →
((visitedExpr : Lean.ExprSet) →
(result : Array Lean.MVarId) → motive { visitedExpr := visitedExpr, result := result }) →
(t : Lean.CollectMVars.State) → motive t | null | false |
SemidirectProduct.rightHom_eq_right | Mathlib.GroupTheory.SemidirectProduct | ∀ {N : Type u_1} {G : Type u_2} [inst : Group N] [inst_1 : Group G] {φ : G →* MulAut N},
⇑SemidirectProduct.rightHom = SemidirectProduct.right | null | true |
AlgebraicGeometry.StructureSheaf.sectionsSubmodule._proof_5 | Mathlib.AlgebraicGeometry.StructureSheaf | ∀ {R : Type u_1} [inst : CommRing R], IsScalarTower R R R | null | false |
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