name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M | docString stringlengths 0 11.5k ⌀ | allowCompletion bool 2
classes |
|---|---|---|---|---|
CategoryTheory.Functor.preservesLimit_cospan_of_mem_presieve | Mathlib.CategoryTheory.Sites.Precoverage | ∀ {C : Type u_1} {D : Type u_2} {inst : CategoryTheory.Category.{v_1, u_1} C}
{inst_1 : CategoryTheory.Category.{v_2, u_2} D} {F : CategoryTheory.Functor C D} {X : C}
(R : CategoryTheory.Presieve X) [self : F.PreservesPairwisePullbacks R] ⦃Y Z : C⦄ ⦃f : Y ⟶ X⦄ ⦃g : Z ⟶ X⦄,
R f → R g → CategoryTheory.Limits.Preser... | **Alias** of `CategoryTheory.Functor.PreservesPairwisePullbacks.preservesLimit`. | true |
MonomialOrder.lex._proof_3 | Mathlib.Data.Finsupp.MonomialOrder | ∀ {σ : Type u_1} (a b : σ →₀ ℕ), toLex (a + b) = toLex a + toLex b | null | false |
CategoryTheory.Arrow.isoOfNatIso._proof_2 | Mathlib.CategoryTheory.Comma.Arrow | ∀ {C : Type u_4} {D : Type u_2} [inst : CategoryTheory.Category.{u_3, u_4} C]
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] {F G : CategoryTheory.Functor C D} (e : F ≅ G)
(f : CategoryTheory.Arrow C),
CategoryTheory.CategoryStruct.comp (e.app f.left).hom (G.mapArrow.obj f).hom =
CategoryTheory.CategoryStruc... | null | false |
Lean.Parser.Tactic.revert | Init.Tactics | Lean.ParserDescr | `revert x...` is the inverse of `intro x...`: it moves the given hypotheses
into the main goal's target type.
| true |
_private.Mathlib.Analysis.SpecialFunctions.BinaryEntropy.0.Real.strictConcaveOn_qaryEntropy._simp_1_1 | Mathlib.Analysis.SpecialFunctions.BinaryEntropy | ∀ {α : Type u_1} [inst : Preorder α] {a b x : α}, (x ∈ Set.Ioo a b) = (a < x ∧ x < b) | null | false |
SetLike.GradeZero.coe_intCast | Mathlib.Algebra.DirectSum.Internal | ∀ {ι : Type u_1} {σ : Type u_2} {R : Type u_4} [inst : Ring R] [inst_1 : AddMonoid ι] [inst_2 : SetLike σ R]
[inst_3 : AddSubgroupClass σ R] (A : ι → σ) [inst_4 : SetLike.GradedMonoid A] (z : ℤ), ↑↑z = ↑z | null | true |
CategoryTheory.Adjunction.homEquiv | Mathlib.CategoryTheory.Adjunction.Basic | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{D : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
{F : CategoryTheory.Functor C D} →
{G : CategoryTheory.Functor D C} → (F ⊣ G) → (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y) | The hom set equivalence associated to an adjunction. | true |
Filter.Realizer.ctorIdx | Mathlib.Data.Analysis.Filter | {α : Type u_1} → {f : Filter α} → f.Realizer → ℕ | null | false |
ValuationSubring.mem_ofSubring._simp_1 | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] (R : Subring K) (hR : ∀ (x : K), x ∈ R ∨ x⁻¹ ∈ R) (x : K),
(x ∈ ValuationSubring.ofSubring R hR) = (x ∈ R) | null | false |
Lean.Doc.Part.brecOn_1 | Lean.DocString.Types | {i : Type u} →
{b : Type v} →
{p : Type w} →
{motive_1 : Lean.Doc.Part i b p → Sort u_1} →
{motive_2 : Array (Lean.Doc.Part i b p) → Sort u_1} →
{motive_3 : List (Lean.Doc.Part i b p) → Sort u_1} →
(t : Array (Lean.Doc.Part i b p)) →
((t : Lean.Doc.Part i b p) → t... | null | false |
CategoryTheory.IsAddMonHom.addMonoidHom | Mathlib.CategoryTheory.Monoidal.Cartesian.Mon | {C : Type u_1} →
[inst : CategoryTheory.Category.{v, u_1} C] →
[inst_1 : CategoryTheory.CartesianMonoidalCategory C] →
{M N : C} →
[inst_2 : CategoryTheory.AddMonObj M] →
[inst_3 : CategoryTheory.AddMonObj N] →
(f : M ⟶ N) → [CategoryTheory.IsAddMonHom f] → (X : C) → (X ⟶ M) →+... | An additive monoid morphism `f : M ⟶ N` induces an additive monoid homomorphism
`M(X) →+ N(X)` for every `X`. | true |
Representation.FiniteCyclicGroup.coinvariantsEquiv | Mathlib.RepresentationTheory.Homological.FiniteCyclic | {k : Type u_1} →
{G : Type u_2} →
[inst : CommRing k] →
[inst_1 : Group G] →
{V : Type u_4} →
[inst_2 : AddCommGroup V] →
[inst_3 : Module k V] →
(ρ : Representation k G V) →
(g : G) →
[Fintype G] →
(∀ (x : G),... | Given a finite cyclic group `G` generated by `g` and a `G` representation `(V, ρ)`, `V_G` is
isomorphic to `V ⧸ Im(ρ(g - 1))`. | true |
TensorProduct.AlgebraTensorModule.lTensor_comp | Mathlib.LinearAlgebra.TensorProduct.Tower | ∀ {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} {Q' : Type uQ'} [inst : CommSemiring R]
[inst_1 : Semiring A] [inst_2 : Algebra R A] [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M]
[inst_6 : IsScalarTower R A M] [inst_7 : AddCommMonoid N] [inst_8 : Module R N] [inst_9 ... | null | true |
String.validFor_mkIterator | Batteries.Data.String.Lemmas | ∀ (s : String), String.Legacy.Iterator.ValidFor [] s.toList (String.Legacy.mkIterator s) | null | true |
AlgebraicGeometry.Scheme.instOverSpecResidueField | Mathlib.AlgebraicGeometry.ResidueField | {X : AlgebraicGeometry.Scheme} → (x : ↥X) → (AlgebraicGeometry.Spec (X.residueField x)).Over X | null | true |
HomologicalComplex.restriction.sc'Iso_hom_τ₃ | Mathlib.Algebra.Homology.Embedding.RestrictionHomology | ∀ {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3}
[inst : CategoryTheory.Category.{v_1, u_3} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(K : HomologicalComplex C c') (e : c.Embedding c') [inst_2 : e.IsRelIff] (i j k : ι) {i' j' k' : ι'}
(hi' : e.f i = i') (hj'... | null | true |
DMatrix.instUnique | Mathlib.Data.Matrix.DMatrix | {m : Type u_1} → {n : Type u_2} → {α : m → n → Type v} → [(i : m) → (j : n) → Unique (α i j)] → Unique (DMatrix m n α) | null | true |
Parser.Attr.qify_simps | Mathlib.Tactic.Attr.Register | Lean.ParserDescr | The simpset `qify_simps` is used by the tactic `qify` to move expressions from `ℕ` or `ℤ` to `ℚ`
which gives a well-behaved division. | true |
List.set_eq_modify | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} (a : α) (n : ℕ) (l : List α), l.set n a = l.modify n fun x => a | null | true |
CongruenceSubgroup.Gamma1_mem._simp_1 | Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | ∀ (N : ℕ) (A : Matrix.SpecialLinearGroup (Fin 2) ℤ),
(A ∈ CongruenceSubgroup.Gamma1 N) = (↑(↑A 0 0) = 1 ∧ ↑(↑A 1 1) = 1 ∧ ↑(↑A 1 0) = 0) | null | false |
_private.Std.Data.DTreeMap.Internal.Lemmas.0.Std.DTreeMap.Internal.Impl.Equiv.forM_eq._simp_1_2 | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {a b : α}, (compare a b ≠ Ordering.eq) = ((a == b) = false) | null | false |
isAddRegular_toColex | Mathlib.Algebra.Order.Group.Synonym | ∀ {α : Type u_1} [inst : AddMonoid α] {a : α}, IsAddRegular (toColex a) ↔ IsAddRegular a | null | true |
_private.Mathlib.CategoryTheory.Comma.Over.Basic.0.CategoryTheory.Over.isRightAdjoint_post.match_1 | Mathlib.CategoryTheory.Comma.Over.Basic | ∀ {T : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} T] {D : Type u_4}
[inst_1 : CategoryTheory.Category.{u_2, u_4} D] {G : CategoryTheory.Functor D T} (motive : G.IsRightAdjoint → Prop)
(x : G.IsRightAdjoint), (∀ (F : CategoryTheory.Functor T D) (a : F ⊣ G), motive ⋯) → motive x | null | false |
MeasureTheory.JordanDecomposition.casesOn | Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {α : Type u_2} →
[inst : MeasurableSpace α] →
{motive : MeasureTheory.JordanDecomposition α → Sort u} →
(t : MeasureTheory.JordanDecomposition α) →
((posPart negPart : MeasureTheory.Measure α) →
[posPart_finite : MeasureTheory.IsFiniteMeasure posPart] →
[negPart_finite : Me... | null | false |
RingEquiv.nonUnitalSubsemiringCongr._proof_1 | Mathlib.RingTheory.NonUnitalSubsemiring.Basic | ∀ {R : Type u_1} [inst : NonUnitalNonAssocSemiring R] {s t : NonUnitalSubsemiring R}, s = t → ↑s = ↑t | null | false |
AffineMap.finrank_eq | Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | ∀ {R : Type u_1} {S : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_5} [inst : Ring R] [inst_1 : Ring S]
[inst_2 : AddCommGroup V] [inst_3 : Module R V] [Module.Finite R V] [Module.Free R V] [inst_6 : AddTorsor V P]
[inst_7 : AddCommGroup W] [inst_8 : Module R W] [inst_9 : Module S W] [Module.Finite S W]
[i... | null | true |
StrictMonoOn.union | Mathlib.Order.Monotone.Union | ∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : Preorder β] {f : α → β} {s t : Set α} {c : α},
StrictMonoOn f s → StrictMonoOn f t → IsGreatest s c → IsLeast t c → StrictMonoOn f (s ∪ t) | If `f` is strictly monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly monotone on `s ∪ t` | true |
PerfectionMap.lift._proof_4 | Mathlib.RingTheory.Perfection | ∀ (p : ℕ) [inst : Fact (Nat.Prime p)] (R : Type u_1) [inst_1 : CommSemiring R] [inst_2 : CharP R p]
[inst_3 : PerfectRing R p] (S : Type u_2) [inst_4 : CommSemiring S] [inst_5 : CharP S p] (P : Type u_3)
[inst_6 : CommSemiring P] [inst_7 : CharP P p] [inst_8 : PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π)
... | null | false |
Polynomial.isMonicOfDegree_X_add_one | Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | ∀ {R : Type u_1} [inst : Semiring R] [Nontrivial R] (r : R), (Polynomial.X + Polynomial.C r).IsMonicOfDegree 1 | null | true |
AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map | Mathlib.Geometry.RingedSpace.LocallyRingedSpace | ∀ {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat}
(h : Topology.IsOpenEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) {X_1 Y : (TopologicalSpace.Opens ↑U)ᵒᵖ}
(f_1 : X_1 ⟶ Y), (X.restrict h).presheaf.map f_1 = X.presheaf.map (h.functor.map f_1.unop).op | null | true |
SkewPolynomial.support_C_mul_X_pow_subset | Mathlib.Algebra.SkewPolynomial.Basic | ∀ {R : Type u_1} [inst : Semiring R] [inst_1 : MulSemiringAction (Multiplicative ℕ) R] (n : ℕ) (c : R),
(SkewPolynomial.C c * SkewPolynomial.X ^ n).support ⊆ {n} | null | true |
Lean.Meta.TransparencyMode.noConfusion | Init.MetaTypes | {P : Sort v✝} → {x y : Lean.Meta.TransparencyMode} → x = y → Lean.Meta.TransparencyMode.noConfusionType P x y | null | false |
FiberBundleCore.Index | Mathlib.Topology.FiberBundle.Basic | {ι : Type u_1} →
{B : Type u_2} →
{F : Type u_3} → [inst : TopologicalSpace B] → [inst_1 : TopologicalSpace F] → FiberBundleCore ι B F → Type u_1 | The index set of a fiber bundle core, as a convenience function for dot notation | true |
Lean.Meta.Grind.instInhabitedTheorems.default | Lean.Meta.Tactic.Grind.Theorems | {α : Type} → Lean.Meta.Grind.Theorems α | null | true |
CategoryTheory.ShortComplex.ShortExact.singleδ.eq_1 | Mathlib.Algebra.Homology.DerivedCategory.SingleTriangle | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
[inst_2 : HasDerivedCategory C] {S : CategoryTheory.ShortComplex C} (hS : S.ShortExact),
hS.singleδ =
CategoryTheory.CategoryStruct.comp
(((CategoryTheory.SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso
... | null | true |
FinBoolAlg.noConfusion | Mathlib.Order.Category.FinBoolAlg | {P : Sort u} → {t t' : FinBoolAlg} → t = t' → FinBoolAlg.noConfusionType P t t' | null | false |
CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply | Mathlib.Algebra.Homology.ShortComplex.ModuleCat | ∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)) {Z : ModuleCat R}
(h : S.moduleCatLeftHomologyData.H ⟶ Z) (x : ↑S.cycles),
(CategoryTheory.ConcreteCategory.hom h)
((CategoryTheory.ConcreteCategory.hom S.moduleCatHomologyIso.hom)
((CategoryTheory.ConcreteCategory.hom S.ho... | null | true |
Plausible.InjectiveFunction.shrink._proof_5 | Mathlib.Testing.Plausible.Functions | ∀ {α : Type} (xs' ys' : List α),
xs'.Perm ys' →
xs'.length ≤ ys'.length →
ys'.length ≤ xs'.length →
(List.map Sigma.fst (List.map Prod.toSigma (xs'.zip ys'))).Perm
(List.map Sigma.snd (List.map Prod.toSigma (xs'.zip ys'))) | null | false |
BooleanSubalgebra.instCompleteLattice._proof_4 | Mathlib.Order.BooleanSubalgebra | ∀ {α : Type u_1} [inst : BooleanAlgebra α] (_L _M : BooleanSubalgebra α) (_a : α), _a ∈ ↑_L ∧ _a ∈ ↑_M → _a ∈ ↑_M | null | false |
_private.Mathlib.Topology.ShrinkingLemma.0.ShrinkingLemma.PartialRefinement.chainSup._simp_8 | Mathlib.Topology.ShrinkingLemma | ∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i | null | false |
Quiver.symmetrifyStar | Mathlib.Combinatorics.Quiver.Covering | {U : Type u_1} →
[inst : Quiver U] → (u : U) → Quiver.Star (Quiver.Symmetrify.of.obj u) ≃ Quiver.Star u ⊕ Quiver.Costar u | The star of the symmetrification of a quiver at a vertex `u` is equivalent to the sum of the
star and the costar at `u` in the original quiver. | true |
CategoryTheory.Comonad.forget_creates_limits_of_comonad_preserves | Mathlib.CategoryTheory.Monad.Limits | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : Type u} [inst_1 : CategoryTheory.Category.{v, u} J]
{T : CategoryTheory.Comonad C} [CategoryTheory.Limits.PreservesLimitsOfShape J T.toFunctor]
(D : CategoryTheory.Functor J T.Coalgebra) [CategoryTheory.Limits.HasLimit (D.comp T.forget)],
CategoryTh... | For `D : J ⥤ Coalgebra T`, `D ⋙ forget T` has a limit, then `D` has a limit provided limits
of shape `J` are preserved by `T`.
| true |
List.SortedLT.map_toDual | Mathlib.Data.List.Sort | ∀ {α : Type u_1} [inst : Preorder α] {l : List α}, (List.map (⇑OrderDual.toDual) l).SortedLT → l.SortedGT | **Alias** of the forward direction of `List.sortedLT_map_toDual`. | true |
_private.Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne.0.SimplexCategory.σ_comp_toMk₁_of_lt._simp_1_5 | Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | ∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a < b) = (b ≤ a) | null | false |
Nat.findGreatest_eq_zero_iff | Mathlib.Data.Nat.Find | ∀ {k : ℕ} {P : ℕ → Prop} [inst : DecidablePred P], Nat.findGreatest P k = 0 ↔ ∀ ⦃n : ℕ⦄, 0 < n → n ≤ k → ¬P n | null | true |
CentroidHom.centerToCentroidCenter | Mathlib.Algebra.Ring.CentroidHom | {α : Type u_5} →
[inst : NonUnitalNonAssocSemiring α] → ↥(NonUnitalSubsemiring.center α) →ₙ+* ↥(Subsemiring.center (CentroidHom α)) | The canonical homomorphism from the center into the center of the centroid | true |
LieEquiv.ofSubalgebras_apply | Mathlib.Algebra.Lie.Subalgebra | ∀ {R : Type u} {L₁ : Type v} {L₂ : Type w} [inst : CommRing R] [inst_1 : LieRing L₁] [inst_2 : LieRing L₂]
[inst_3 : LieAlgebra R L₁] [inst_4 : LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂)
(e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') (x : ↥L₁'),
↑((LieEquiv.ofSubalgebra... | null | true |
_private.Std.Data.DHashMap.Internal.Raw.0.Std.DHashMap.Raw.diff.eq_1 | Std.Data.DHashMap.Internal.Raw | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] [inst_1 : Hashable α] (m₁ m₂ : Std.DHashMap.Raw α β),
m₁.diff m₂ =
if h₁ : 0 < m₁.buckets.size then
if h₂ : 0 < m₂.buckets.size then ↑(Std.DHashMap.Internal.Raw₀.diff ⟨m₁, h₁⟩ ⟨m₂, h₂⟩) else m₁
else m₂ | null | true |
CategoryTheory.isFinitelyPresentable_iff_preservesFilteredColimits | Mathlib.CategoryTheory.Presentable.Finite | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X : C},
CategoryTheory.IsFinitelyPresentable X ↔
CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.coyoneda.obj (Opposite.op X)) | null | true |
isPreirreducible_iff_isClosed_union_isClosed | Mathlib.Topology.Irreducible | ∀ {X : Type u_1} [inst : TopologicalSpace X] {s : Set X},
IsPreirreducible s ↔ ∀ (z₁ z₂ : Set X), IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂ | A set is preirreducible if and only if
for every cover by two closed sets, it is contained in one of the two covering sets. | true |
IntermediateField.LinearDisjoint.symm | Mathlib.FieldTheory.LinearDisjoint | ∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {A B : IntermediateField F E},
A.LinearDisjoint ↥B → B.LinearDisjoint ↥A | Linear disjointness is symmetric. | true |
instModuleTensorProductMop._proof_1 | Mathlib.Algebra.Azumaya.Defs | ∀ (R : Type u_1) (A : Type u_2) [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A],
IsScalarTower R Aᵐᵒᵖ A | null | false |
Std.Async.System.Environment.get? | Std.Async.System | Std.Async.System.Environment → String → Option String | null | true |
eventuallyMeasurableSpace._proof_1 | Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable | ∀ {α : Type u_1} (m : MeasurableSpace α) (l : Filter α), ∃ t, MeasurableSet t ∧ ∅ =ᶠ[l] t | null | false |
Lean.Elab.Tactic.RCases.RCasesPatt.one | Lean.Elab.Tactic.RCases | Lean.Syntax → Lean.Name → Lean.Elab.Tactic.RCases.RCasesPatt | A named pattern like `foo` | true |
CategoryTheory.Limits.ChosenPullback₃.w₁_assoc | Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ S : C} {f₁ : X₁ ⟶ S} {f₂ : X₂ ⟶ S} {f₃ : X₃ ⟶ S}
{h₁₂ : CategoryTheory.Limits.ChosenPullback f₁ f₂} {h₂₃ : CategoryTheory.Limits.ChosenPullback f₂ f₃}
{h₁₃ : CategoryTheory.Limits.ChosenPullback f₁ f₃} (h : CategoryTheory.Limits.ChosenPullback₃ h₁₂ ... | null | true |
AlgebraicGeometry.Scheme.mem_zeroLocus_iff._simp_1 | Mathlib.AlgebraicGeometry.Scheme | ∀ (X : AlgebraicGeometry.Scheme) {U : X.Opens} (s : Set ↑(X.presheaf.obj (Opposite.op U))) (x : ↥X),
(x ∈ X.zeroLocus s) = ∀ f ∈ s, x ∉ X.basicOpen f | null | false |
Mathlib.Tactic.Coherence.LiftHom.noConfusion | Mathlib.Tactic.CategoryTheory.Coherence | {P : Sort u_1} →
{C : Type u} →
{inst : CategoryTheory.Category.{v, u} C} →
{X Y : C} →
{inst_1 : Mathlib.Tactic.Coherence.LiftObj X} →
{inst_2 : Mathlib.Tactic.Coherence.LiftObj Y} →
{f : X ⟶ Y} →
{t : Mathlib.Tactic.Coherence.LiftHom f} →
{C' : T... | null | false |
CategoryTheory.Functor.mapAddGrpFunctor._proof_8 | Mathlib.CategoryTheory.Monoidal.Grp | ∀ {C : Type u_3} [inst : CategoryTheory.Category.{u_1, u_3} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
{D : Type u_4} [inst_2 : CategoryTheory.Category.{u_2, u_4} D] [inst_3 : CategoryTheory.CartesianMonoidalCategory D]
(X : C ⥤ₗ D),
{ app := fun A => CategoryTheory.AddGrp.homMk'' ((CategoryTheory.C... | null | false |
Lean.Meta.ZetaUnusedMode.ctorElim | Lean.Meta.HaveTelescope | {motive : Lean.Meta.ZetaUnusedMode → Sort u} →
(ctorIdx : ℕ) →
(t : Lean.Meta.ZetaUnusedMode) → ctorIdx = t.ctorIdx → Lean.Meta.ZetaUnusedMode.ctorElimType ctorIdx → motive t | null | false |
List.find?_append | Init.Data.List.Impl | ∀ {α : Type u_1} {p : α → Bool} {xs ys : List α}, List.find? p (xs ++ ys) = (List.find? p xs).or (List.find? p ys) | null | true |
StandardSubspace.mk._flat_ctor | Mathlib.Analysis.InnerProductSpace.StandardSubspace | {H : Type u_1} →
[inst : NormedAddCommGroup H] →
[inst_1 : InnerProductSpace ℂ H] →
(toClosedSubmodule : ClosedSubmodule ℝ H) →
toClosedSubmodule ⊓ toClosedSubmodule.mulI = ⊥ →
toClosedSubmodule ⊔ toClosedSubmodule.mulI = ⊤ → StandardSubspace H | null | false |
IntermediateField.instAlgebraSubtypeMemAdjoinSingletonSetCoeRingHomAlgebraMap._proof_1 | Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | ∀ {A : Type u_2} {B : Type u_1} [inst : Field A] [inst_1 : Field B] [inst_2 : Algebra A B],
SubsemiringClass (IntermediateField A B) B | null | false |
CategoryTheory.Bicategory.Equivalence.mk.inj | Mathlib.CategoryTheory.Bicategory.Adjunction.Basic | ∀ {B : Type u} {inst : CategoryTheory.Bicategory B} {a b : B} {hom : a ⟶ b} {inv : b ⟶ a}
{unit : CategoryTheory.CategoryStruct.id a ≅ CategoryTheory.CategoryStruct.comp hom inv}
{counit : CategoryTheory.CategoryStruct.comp inv hom ≅ CategoryTheory.CategoryStruct.id b}
{left_triangle :
autoParam
(Catego... | null | true |
Ideal.IsTwoSided.mul_one | Mathlib.RingTheory.Ideal.Operations | ∀ {R : Type u} [inst : Semiring R] {I : Ideal R} [I.IsTwoSided], I * 1 = I | null | true |
_private.Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis.0.AlgebraicIndependent.matroid_spanning_iff_of_subsingleton._simp_1_2 | Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | ∀ {α : Type u_2} (M : Matroid α) (S : Set α), M.Spanning S = (M.closure S = M.E ∧ S ⊆ M.E) | null | false |
MeasureTheory.Lp.constₗ | Mathlib.MeasureTheory.Function.LpSpace.Indicator | {α : Type u_1} →
{E : Type u_2} →
{m : MeasurableSpace α} →
(p : ENNReal) →
(μ : MeasureTheory.Measure α) →
[inst : NormedAddCommGroup E] →
[MeasureTheory.IsFiniteMeasure μ] →
(𝕜 : Type u_3) →
[inst_2 : NormedRing 𝕜] →
[inst_3 :... | `MeasureTheory.Lp.const` as a `LinearMap`. | true |
SheafOfModules.hasLimitsOfSize | Mathlib.Algebra.Category.ModuleCat.Sheaf.Limits | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C}
(R : CategoryTheory.Sheaf J RingCat),
CategoryTheory.Limits.HasLimitsOfSize.{v₂, v, max u₁ v, max (max (max (v + 1) u) u₁) v₁} (SheafOfModules R) | null | true |
LinearPMap | Mathlib.LinearAlgebra.LinearPMap | {R : Type u_1} →
{S : Type u_2} →
[inst : Ring R] →
[inst_1 : Ring S] →
(R →+* S) →
(E : Type u_3) →
[inst_2 : AddCommGroup E] →
[Module R E] → (F : Type u_4) → [inst : AddCommGroup F] → [Module S F] → Type (max u_3 u_4) | A `LinearPMap σ E F` or `E →ₛₗ.[σ] F` is a (semi)linear map from a submodule of `E` to `F`. | true |
Mathlib.Tactic.DepRewrite.Conv.depRw | Mathlib.Tactic.DepRewrite | Lean.ParserDescr | `rw!` is like `rewrite!`, but also cleans up introduced refl-casts after every substitution.
It is available as an ordinary tactic and a `conv` tactic.
| true |
_private.Lean.Meta.Tactic.Grind.EMatchAction.0.Lean.Meta.Grind.Action.CollectState.collectedThms._default | Lean.Meta.Tactic.Grind.EMatchAction | Std.HashSet (Lean.Meta.Grind.Origin × Lean.Meta.Grind.EMatchTheoremKind) | null | false |
_private.Mathlib.Combinatorics.Additive.FreimanHom.0.Fin.isAddFreimanIso_Iio._simp_1_1 | Mathlib.Combinatorics.Additive.FreimanHom | ∀ {α : Type u_1} [inst : Preorder α] {b x : α}, (x ∈ Set.Iio b) = (x < b) | null | false |
MulArchimedeanOrder.le_def | Mathlib.Algebra.Order.Archimedean.Class | ∀ {M : Type u_1} [inst : Group M] [inst_1 : Lattice M] {a b : MulArchimedeanOrder M},
a ≤ b ↔ ∃ n, |MulArchimedeanOrder.val b|ₘ ≤ |MulArchimedeanOrder.val a|ₘ ^ n | null | true |
ProbabilityTheory.«_aux_Mathlib_Probability_ConditionalProbability___macroRules_ProbabilityTheory_term__[|_]_1» | Mathlib.Probability.ConditionalProbability | Lean.Macro | null | false |
Associated.of_pow_associated_of_prime' | Mathlib.Algebra.GroupWithZero.Associated | ∀ {M : Type u_1} [inst : CommMonoidWithZero M] [IsCancelMulZero M] {p₁ p₂ : M} {k₁ k₂ : ℕ},
Prime p₁ → Prime p₂ → 0 < k₂ → Associated (p₁ ^ k₁) (p₂ ^ k₂) → Associated p₁ p₂ | null | true |
MeasureTheory.Filtration.natural | Mathlib.Probability.Process.Filtration | {Ω : Type u_1} →
{ι : Type u_2} →
{m : MeasurableSpace Ω} →
{β : ι → Type u_3} →
[inst : (i : ι) → TopologicalSpace (β i)] →
[∀ (i : ι), TopologicalSpace.MetrizableSpace (β i)] →
[mβ : (i : ι) → MeasurableSpace (β i)] →
[∀ (i : ι), BorelSpace (β i)] →
... | Given a sequence of functions, the natural filtration is the smallest sequence
of σ-algebras such that the sequence of functions is measurable with respect to
the filtration. | true |
ContinuousLinearMap.reApplyInnerSelf_continuous | Mathlib.Analysis.InnerProductSpace.LinearMap | ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
(T : E →L[𝕜] E), Continuous T.reApplyInnerSelf | null | true |
CategoryTheory.Limits.Cocones.postcomposeId | Mathlib.CategoryTheory.Limits.Cones | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
{C : Type u₃} →
[inst_1 : CategoryTheory.Category.{v₃, u₃} C] →
{F : CategoryTheory.Functor J C} →
CategoryTheory.Limits.Cocone.precompose (CategoryTheory.CategoryStruct.id F) ≅
CategoryTheory.Functor.id (CategoryThe... | **Alias** of `CategoryTheory.Limits.Cocone.precomposeId`.
---
Precomposing by the identity does not change the cocone up to isomorphism. | true |
List.toChunks.go | Batteries.Data.List.Basic | {α : Type u_1} → ℕ → List α → Array α → Array (List α) → List (List α) | Auxliary definition used to define `toChunks`.
`toChunks.go xs acc₁ acc₂` pushes elements into `acc₁` until it reaches size `n`,
then it pushes the resulting list to `acc₂` and continues until `xs` is exhausted. | true |
_private.Mathlib.NumberTheory.Chebyshev.0.Chebyshev.primeCounting_eq_theta_div_log_add_integral._simp_1_7 | Mathlib.NumberTheory.Chebyshev | ∀ {R : Type u_1} [inst : AddMonoidWithOne R] [CharZero R] (n : ℕ), (↑n + 1 = 0) = False | null | false |
LightCondensed.instMonoidalLightCondSetLightCondModFree._aux_10 | Mathlib.Condensed.Light.Monoidal | (R : Type u_1) →
[inst : CommRing R] →
(LightCondensed.free R).obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit LightCondSet) ⟶
CategoryTheory.MonoidalCategoryStruct.tensorUnit (LightCondMod R) | null | false |
CategoryTheory.WithInitial.of.injEq | Mathlib.CategoryTheory.WithTerminal.Basic | ∀ {C : Type u} (a a_1 : C), (CategoryTheory.WithInitial.of a = CategoryTheory.WithInitial.of a_1) = (a = a_1) | null | true |
Path.extend | Mathlib.Topology.Path | {X : Type u_1} → [inst : TopologicalSpace X] → {x y : X} → Path x y → C(ℝ, X) | A continuous map extending a path to `ℝ`, constant before `0` and after `1`. | true |
conditionallyCompleteLatticeOfLatticeOfsSup | Mathlib.Order.ConditionallyCompleteLattice.Defs | (α : Type u_5) →
[H1 : Lattice α] →
[inst : SupSet α] → (∀ (s : Set α), BddAbove s → s.Nonempty → IsLUB s (sSup s)) → ConditionallyCompleteLattice α | A version of `conditionallyCompleteLatticeOfsSup` when we already know that `α` is a lattice.
This should only be used when it is both hard and unnecessary to provide `sInf` explicitly. | true |
Lean.Linter.UnusedVariables.References.casesOn | Lean.Linter.UnusedVariables | {motive : Lean.Linter.UnusedVariables.References → Sort u} →
(t : Lean.Linter.UnusedVariables.References) →
((constDecls : Std.HashSet Lean.Syntax.Range) →
(fvarDefs : Std.HashMap Lean.Syntax.Range Lean.Linter.UnusedVariables.FVarDefinition) →
(fvarUses : Std.HashSet Lean.FVarId) →
(... | null | false |
CategoryTheory.PreZeroHypercover.Hom.mk.injEq | Mathlib.CategoryTheory.Sites.Hypercover.Zero | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {S : C} {E : CategoryTheory.PreZeroHypercover S}
{F : CategoryTheory.PreZeroHypercover S} (s₀ : E.I₀ → F.I₀) (h₀ : (i : E.I₀) → E.X i ⟶ F.X (s₀ i))
(w₀ :
autoParam (∀ (i : E.I₀), CategoryTheory.CategoryStruct.comp (h₀ i) (F.f (s₀ i)) = E.f i)
Catego... | null | true |
Equiv.Set.sumDiffSubset_apply_inl | Mathlib.Logic.Equiv.Set | ∀ {α : Type u_3} {s t : Set α} (h : s ⊆ t) [inst : DecidablePred fun x => x ∈ s] (x : ↑s),
(Equiv.Set.sumDiffSubset h) (Sum.inl x) = Set.inclusion h x | null | true |
Lean.Lsp.PrepareRenameParams._sizeOf_1 | Lean.Data.Lsp.LanguageFeatures | Lean.Lsp.PrepareRenameParams → ℕ | null | false |
_private.Mathlib.Algebra.SkewMonoidAlgebra.Basic.0.SkewMonoidAlgebra.coeff.match_1.eq_1 | Mathlib.Algebra.SkewMonoidAlgebra.Basic | ∀ {k : Type u_1} {G : Type u_2} [inst : AddMonoid k] (motive : SkewMonoidAlgebra k G → Sort u_3) (p : G →₀ k)
(h_1 : (p : G →₀ k) → motive { toFinsupp := p }),
(match { toFinsupp := p } with
| { toFinsupp := p } => h_1 p) =
h_1 p | null | true |
_private.Mathlib.Algebra.Regular.Defs.0.isRegular_iff.match_1_3 | Mathlib.Algebra.Regular.Defs | ∀ {R : Type u_1} [inst : Mul R] {c : R} (motive : IsLeftRegular c ∧ IsRightRegular c → Prop)
(x : IsLeftRegular c ∧ IsRightRegular c), (∀ (h1 : IsLeftRegular c) (h2 : IsRightRegular c), motive ⋯) → motive x | null | false |
Option.forIn'_join._proof_1 | Init.Data.Option.Monadic | ∀ {α : Type u_1} (o : Option (Option α)), ∀ o' ∈ o, ∀ a ∈ o', a ∈ o.join | null | false |
Std.Tactic.BVDecide.BVExpr.Cache.insert._proof_4 | Std.Tactic.BVDecide.Bitblast.BVExpr.Circuit.Impl.Expr | ∀ {aig : Std.Sat.AIG Std.Tactic.BVDecide.BVBit} {w : ℕ} (expr : Std.Tactic.BVDecide.BVExpr w) (refs : aig.RefVec w)
(map : Std.DHashMap Std.Tactic.BVDecide.BVExpr.Cache.Key fun k => Vector Std.Sat.AIG.Fanin k.w),
(∀ {i : ℕ} (k : Std.Tactic.BVDecide.BVExpr.Cache.Key) (h1 : k ∈ map) (h2 : i < k.w),
(map.get k h... | null | false |
CategoryTheory.Limits.kernelFactorThruImage.eq_1 | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C}
(f : X ⟶ Y) [inst_2 : CategoryTheory.Limits.HasKernel f] [inst_3 : CategoryTheory.Limits.HasImage f]
[inst_4 : CategoryTheory.Limits.HasKernel (CategoryTheory.Limits.factorThruImage f)],
Category... | null | true |
_private.BatteriesRecycling.MonadSatisfying.Basic.0.SatisfiesM_ExceptT_eq.match_1_16 | BatteriesRecycling.MonadSatisfying.Basic | ∀ {α ρ : Type u_1} {p : α → Prop} (motive : { a // ∀ (a_1 : α), a = Except.ok a_1 → p a_1 } → Prop)
(h : { a // ∀ (a_1 : α), a = Except.ok a_1 → p a_1 }),
(∀ (a : Except ρ α) (h : ∀ (a_1 : α), a = Except.ok a_1 → p a_1), motive ⟨a, h⟩) → motive h | null | false |
_private.Mathlib.Topology.Connected.Basic.0.IsPreconnected.iUnion_of_reflTransGen._simp_1_1 | Mathlib.Topology.Connected.Basic | ∀ {α : Type u} (x : α), (x ∈ Set.univ) = True | null | false |
_private.Lean.Meta.Tactic.Grind.EMatch.0.Lean.Meta.Grind.EMatch.checkDefEq.match_1 | Lean.Meta.Tactic.Grind.EMatch | (motive : Array Lean.Expr × Array Lean.BinderInfo × Lean.Expr → Sort u_1) →
(x : Array Lean.Expr × Array Lean.BinderInfo × Lean.Expr) →
((fst : Array Lean.Expr) → (fst_1 : Array Lean.BinderInfo) → (rhsExpr : Lean.Expr) → motive (fst, fst_1, rhsExpr)) →
motive x | null | false |
_private.Lean.OriginalConstKind.0.Lean.wasOriginallyTheorem.match_1 | Lean.OriginalConstKind | (motive : Lean.ConstantKind → Sort u_1) →
(x : Lean.ConstantKind) → (Unit → motive Lean.ConstantKind.thm) → ((x : Lean.ConstantKind) → motive x) → motive x | null | false |
CategoryTheory.ShortComplex.rightHomologyIso_hom_comp_homologyι_assoc | Mathlib.Algebra.Homology.ShortComplex.Homology | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : S.HasHomology] {Z : C} (h : S.opcycles ⟶ Z),
CategoryTheory.CategoryStruct.comp S.rightHomologyIso.hom (CategoryTheory.CategoryStruct.comp S.homologyι h) =
C... | null | true |
Equiv.optionCongr_apply | Mathlib.Logic.Equiv.Option | ∀ {α : Type u_1} {β : Type u_2} (e : α ≃ β) (a : Option α), e.optionCongr a = Option.map (⇑e) a | null | true |
_private.Lean.Meta.Tactic.Cbv.BuiltinCbvSimprocs.String.0.Lean.Meta.Tactic.Cbv.simpStringAppend.match_1 | Lean.Meta.Tactic.Cbv.BuiltinCbvSimprocs.String | (motive : Option String → Sort u_1) →
(x : Option String) → ((b : String) → motive (some b)) → ((x : Option String) → motive x) → motive x | null | false |
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