name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
SSet.RelativeMorphism.Homotopy.h₀._autoParam | Mathlib.AlgebraicTopology.SimplicialSet.RelativeMorphism | Lean.Syntax |
List.ranges._sunfold | Mathlib.Data.List.Range | List ℕ → List (List ℕ) |
Turing.PartrecToTM2.contSupp.eq_3 | Mathlib.Computability.TMToPartrec | ∀ (f : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont'),
Turing.PartrecToTM2.contSupp (Turing.PartrecToTM2.Cont'.comp f k) =
Turing.PartrecToTM2.codeSupp' f k ∪ Turing.PartrecToTM2.contSupp k |
_private.Lean.Elab.Extra.0.Lean.Elab.Term.Op.Tree.unop.inj | Lean.Elab.Extra | ∀ {ref : Lean.Syntax} {f : Lean.Expr} {arg : Lean.Elab.Term.Op.Tree✝} {ref_1 : Lean.Syntax} {f_1 : Lean.Expr}
{arg_1 : Lean.Elab.Term.Op.Tree✝¹},
Lean.Elab.Term.Op.Tree.unop✝ ref f arg = Lean.Elab.Term.Op.Tree.unop✝¹ ref_1 f_1 arg_1 →
ref = ref_1 ∧ f = f_1 ∧ arg = arg_1 |
Stream'.WSeq.productive_tail | Mathlib.Data.WSeq.Productive | ∀ {α : Type u} (s : Stream'.WSeq α) [s.Productive], s.tail.Productive |
Matrix.conjTranspose_list_sum | Mathlib.LinearAlgebra.Matrix.ConjTranspose | ∀ {m : Type u_2} {n : Type u_3} {α : Type v} [inst : AddMonoid α] [inst_1 : StarAddMonoid α] (l : List (Matrix m n α)),
l.sum.conjTranspose = (List.map Matrix.conjTranspose l).sum |
SimpleGraph.ConnectedComponent.connected_toSubgraph | Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | ∀ {V : Type u} {G : SimpleGraph V} (C : G.ConnectedComponent), C.toSubgraph.Connected |
WeierstrassCurve.Jacobian.Point.instAddCommGroup._proof_8 | Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | ∀ {F : Type u_1} [inst : Field F] {W : WeierstrassCurve.Jacobian F} (n : ℕ) (a : W.Point),
zsmulRec nsmulRec (↑n.succ) a = zsmulRec nsmulRec (↑n) a + a |
Fin.succ_lt_succ_iff._simp_1 | Init.Data.Fin.Lemmas | ∀ {n : ℕ} {a b : Fin n}, (a.succ < b.succ) = (a < b) |
Std.TreeMap.Raw.contains_ofList | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} [Std.TransCmp cmp] [inst : BEq α] [Std.LawfulBEqCmp cmp]
{l : List (α × β)} {k : α}, (Std.TreeMap.Raw.ofList l cmp).contains k = (List.map Prod.fst l).contains k |
CategoryTheory.CartesianClosed.curry_id_eq_coev | Mathlib.CategoryTheory.Monoidal.Closed.Cartesian | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.MonoidalCategory C] (A X : C)
[inst_2 : CategoryTheory.Closed A],
CategoryTheory.MonoidalClosed.curry
(CategoryTheory.CategoryStruct.id
(CategoryTheory.MonoidalCategoryStruct.tensorObj A ((CategoryTheory.Functor.id C).obj X))) =
(CategoryTheory.ihom.coev A).app X |
FirstOrder.Language.ElementaryEmbedding.noConfusion | Mathlib.ModelTheory.ElementaryMaps | {P : Sort u} →
{L : FirstOrder.Language} →
{M : Type u_1} →
{N : Type u_2} →
{inst : L.Structure M} →
{inst_1 : L.Structure N} →
{t : L.ElementaryEmbedding M N} →
{L' : FirstOrder.Language} →
{M' : Type u_1} →
{N' : Type u_2} →
{inst' : L'.Structure M'} →
{inst'_1 : L'.Structure N'} →
{t' : L'.ElementaryEmbedding M' N'} →
L = L' →
M = M' →
N = N' →
inst ≍ inst' →
inst_1 ≍ inst'_1 →
t ≍ t' → FirstOrder.Language.ElementaryEmbedding.noConfusionType P t t' |
antivaryOn_neg | Mathlib.Algebra.Order.Monovary | ∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : AddCommGroup α] [inst_1 : PartialOrder α] [IsOrderedAddMonoid α]
[inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] {s : Set ι} {f : ι → α} {g : ι → β},
AntivaryOn (-f) (-g) s ↔ AntivaryOn f g s |
Ordnode.splitMax | Mathlib.Data.Ordmap.Ordnode | {α : Type u_1} → Ordnode α → Option (Ordnode α × α) |
CategoryTheory.Comma.leftIso | Mathlib.CategoryTheory.Comma.Basic | {A : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} A] →
{B : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} B] →
{T : Type u₃} →
[inst_2 : CategoryTheory.Category.{v₃, u₃} T] →
{L₁ : CategoryTheory.Functor A T} →
{R₁ : CategoryTheory.Functor B T} → {X Y : CategoryTheory.Comma L₁ R₁} → (X ≅ Y) → (X.left ≅ Y.left) |
Matroid.ext_spanning | Mathlib.Combinatorics.Matroid.Closure | ∀ {α : Type u_2} {M M' : Matroid α}, M.E = M'.E → (∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S) → M = M' |
Nat.Linear.PolyCnstr.mk | Init.Data.Nat.Linear | Bool → Nat.Linear.Poly → Nat.Linear.Poly → Nat.Linear.PolyCnstr |
_private.Std.Time.Format.Basic.0.Std.Time.instReprFormatPart.repr.match_1 | Std.Time.Format.Basic | (motive : Std.Time.FormatPart → Sort u_1) →
(x : Std.Time.FormatPart) →
((a : String) → motive (Std.Time.FormatPart.string a)) →
((a : Std.Time.Modifier) → motive (Std.Time.FormatPart.modifier a)) → motive x |
Function.locallyFinsuppWithin.instMaxOfSemilatticeSup | Mathlib.Topology.LocallyFinsupp | {X : Type u_1} →
[inst : TopologicalSpace X] →
{U : Set X} → {Y : Type u_2} → [SemilatticeSup Y] → [inst_2 : Zero Y] → Max (Function.locallyFinsuppWithin U Y) |
WithCStarModule.inner_single_right | Mathlib.Analysis.CStarAlgebra.Module.Constructions | ∀ {A : Type u_1} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] {ι : Type u_2} {E : ι → Type u_3}
[inst_2 : Fintype ι] [inst_3 : (i : ι) → NormedAddCommGroup (E i)] [inst_4 : (i : ι) → Module ℂ (E i)]
[inst_5 : (i : ι) → SMul A (E i)] [inst_6 : (i : ι) → CStarModule A (E i)] [inst_7 : StarOrderedRing A]
[inst_8 : DecidableEq ι] (x : WithCStarModule A ((i : ι) → E i)) {i : ι} (y : E i),
inner A x ((WithCStarModule.equiv A ((i : ι) → E i)).symm (Pi.single i y)) = inner A (x i) y |
Aesop.MVarClusterData.goals | Aesop.Tree.Data | {Goal Rapp : Type} → Aesop.MVarClusterData Goal Rapp → Array (IO.Ref Goal) |
ProofWidgets.CheckRequestResponse.noConfusion | ProofWidgets.Cancellable | {P : Sort u} →
{t t' : ProofWidgets.CheckRequestResponse} → t = t' → ProofWidgets.CheckRequestResponse.noConfusionType P t t' |
_private.Init.Data.Vector.Basic.0.Vector.mapM.go._unary._proof_4 | Init.Data.Vector.Basic | ∀ {n : ℕ}, ∀ k < n, k + 1 ≤ n |
_private.Lean.Server.Completion.CompletionItemCompression.0.Lean.Lsp.ResolvableCompletionList.compressEditFast | Lean.Server.Completion.CompletionItemCompression | String → Lean.Lsp.InsertReplaceEdit → String |
_private.Mathlib.Combinatorics.Matroid.Circuit.0.Matroid.fundCircuit_eq_sInter._simp_1_2 | Mathlib.Combinatorics.Matroid.Circuit | ∀ {α : Type u} {a : α} {s : Set α}, ({a} ⊆ s) = (a ∈ s) |
ULift.mul | Mathlib.Algebra.Group.ULift | {α : Type u} → [Mul α] → Mul (ULift.{u_1, u} α) |
LowerSemicontinuousWithinAt.le_liminf | Mathlib.Topology.Semicontinuity.Basic | ∀ {α : Type u_1} [inst : TopologicalSpace α] {s : Set α} {x : α} {γ : Type u_4} [inst_1 : CompleteLinearOrder γ]
{f : α → γ}, LowerSemicontinuousWithinAt f s x → f x ≤ Filter.liminf f (nhdsWithin x s) |
Int.Nonneg.mul | Init.Data.Int.OfNat | ∀ (a b : ℤ), a ≥ 0 → b ≥ 0 → a * b ≥ 0 |
Finset.Ico_subset_Iio_self | Mathlib.Order.Interval.Finset.Basic | ∀ {α : Type u_2} {a b : α} [inst : Preorder α] [inst_1 : LocallyFiniteOrderBot α] [inst_2 : LocallyFiniteOrder α],
Finset.Ico a b ⊆ Finset.Iio b |
measurable_to_countable' | Mathlib.MeasureTheory.MeasurableSpace.Constructions | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace α] [Countable α] [inst_2 : MeasurableSpace β] {f : β → α},
(∀ (x : α), MeasurableSet (f ⁻¹' {x})) → Measurable f |
«_aux_Mathlib_NumberTheory_Padics_Complex___macroRules_term𝓞_ℂ_[_]_1» | Mathlib.NumberTheory.Padics.Complex | Lean.Macro |
CategoryTheory.MonoidalCategory.selfLeftAction._proof_13 | Mathlib.CategoryTheory.Monoidal.Action.Basic | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.MonoidalCategory C] (c : C)
{c' c'' : C} (f : c' ⟶ c'') (d : C),
CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategoryStruct.whiskerLeft c f) d =
CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator c c' d).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft c (CategoryTheory.MonoidalCategoryStruct.whiskerRight f d))
(CategoryTheory.MonoidalCategoryStruct.associator c c'' d).inv) |
CancelCommMonoidWithZero.mk.sizeOf_spec | Mathlib.Algebra.GroupWithZero.Defs | ∀ {M₀ : Type u_2} [inst : SizeOf M₀] (toCommMonoidWithZero : CommMonoidWithZero M₀)
(toIsLeftCancelMulZero : IsLeftCancelMulZero M₀),
sizeOf { toCommMonoidWithZero := toCommMonoidWithZero, toIsLeftCancelMulZero := toIsLeftCancelMulZero } =
1 + sizeOf toCommMonoidWithZero + sizeOf toIsLeftCancelMulZero |
lipschitzOnWith_iff_norm_inv_mul_le | Mathlib.Analysis.Normed.Group.Uniform | ∀ {E : Type u_2} {F : Type u_3} [inst : SeminormedGroup E] [inst_1 : SeminormedGroup F] {s : Set E} {f : E → F}
{C : NNReal}, LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖(f x)⁻¹ * f y‖ ≤ ↑C * ‖x⁻¹ * y‖ |
BitVec.umulOverflow.eq_1 | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} (x y : BitVec w), x.umulOverflow y = decide (x.toNat * y.toNat ≥ 2 ^ w) |
_private.Mathlib.Data.List.Nodup.0.List.nodup_iff_count_eq_one._proof_1_4 | Mathlib.Data.List.Nodup | ∀ {α : Type u_1} {l : List α} [inst : BEq α] (x : α),
(List.count x l ≤ 1) = ¬(1 ≤ List.count x l → List.count x l = 1) →
List.count x l ≤ 1 → 0 < (List.filter (fun x_1 => x_1 == x) l).length |
_private.Mathlib.RingTheory.HahnSeries.Basic.0.HahnSeries.ofIterate._simp_3 | Mathlib.RingTheory.HahnSeries.Basic | ∀ {α : Type u} {β : Type v} {f : α → β} {s : Set β} {a : α}, (a ∈ f ⁻¹' s) = (f a ∈ s) |
InnerProductGeometry.norm_add_eq_norm_sub_iff_angle_eq_pi_div_two | Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | ∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (x y : V),
‖x + y‖ = ‖x - y‖ ↔ InnerProductGeometry.angle x y = Real.pi / 2 |
CategoryTheory.ShortComplex.leftHomologyIso_hom_naturality_assoc | Mathlib.Algebra.Homology.ShortComplex.Homology | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology] (φ : S₁ ⟶ S₂) {Z : C}
(h : S₂.homology ⟶ Z),
CategoryTheory.CategoryStruct.comp S₁.leftHomologyIso.hom
(CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ) h) =
CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap φ)
(CategoryTheory.CategoryStruct.comp S₂.leftHomologyIso.hom h) |
Lean.Meta.FindSplitImpl.Context.mk | Lean.Meta.Tactic.SplitIf | Lean.ExprSet → Lean.Meta.SplitKind → Lean.Meta.FindSplitImpl.Context |
Multiset.card_eq_countP_add_countP | Mathlib.Data.Multiset.Count | ∀ {α : Type u_1} (p : α → Prop) [inst : DecidablePred p] (s : Multiset α),
s.card = Multiset.countP p s + Multiset.countP (fun x => ¬p x) s |
Nat.and_left_comm | Batteries.Data.Nat.Bitwise | ∀ (x y z : ℕ), x &&& (y &&& z) = y &&& (x &&& z) |
Substring.Raw.Valid.next_stop | Batteries.Data.String.Lemmas | ∀ {s : Substring.Raw}, s.Valid → s.next { byteIdx := s.bsize } = { byteIdx := s.bsize } |
_private.Init.Meta.0.Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_declareSimpLikeTactic_1.match_4 | Init.Meta | (motive : Lean.TSyntax `term × Lean.TSyntax `term × Lean.TSyntax `command → Sort u_1) →
(__discr : Lean.TSyntax `term × Lean.TSyntax `term × Lean.TSyntax `command) →
((kind tkn : Lean.TSyntax `term) → (stx : Lean.TSyntax `command) → motive (kind, tkn, stx)) → motive __discr |
WithBot.insertBot | Mathlib.Order.Interval.Finset.Defs | {α : Type u_1} → Finset α ↪o Finset (WithBot α) |
Finset.preimage_subset | Mathlib.Data.Finset.Preimage | ∀ {α : Type u} {β : Type v} {f : α ↪ β} {s : Finset β} {t : Finset α}, s ⊆ Finset.map f t → s.preimage ⇑f ⋯ ⊆ t |
Lean.IR.ToIR.BuilderState.mk.inj | Lean.Compiler.IR.ToIR | ∀ {vars : Std.HashMap Lean.FVarId Lean.IR.Arg} {joinPoints : Std.HashMap Lean.FVarId Lean.IR.JoinPointId} {nextId : ℕ}
{vars_1 : Std.HashMap Lean.FVarId Lean.IR.Arg} {joinPoints_1 : Std.HashMap Lean.FVarId Lean.IR.JoinPointId}
{nextId_1 : ℕ},
{ vars := vars, joinPoints := joinPoints, nextId := nextId } =
{ vars := vars_1, joinPoints := joinPoints_1, nextId := nextId_1 } →
vars = vars_1 ∧ joinPoints = joinPoints_1 ∧ nextId = nextId_1 |
Algebra.semiringToRing._proof_1 | Mathlib.Algebra.Algebra.Basic | ∀ {A : Type u_1} (R : Type u_2) [inst : CommRing R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (z : ℕ),
(algebraMap R A) ↑↑z = ↑z |
BoundedContinuousFunction.natCast_apply | Mathlib.Topology.ContinuousMap.Bounded.Basic | ∀ {α : Type u} [inst : TopologicalSpace α] {β : Type u_2} [inst_1 : PseudoMetricSpace β] [inst_2 : NatCast β] (n : ℕ)
(x : α), ↑n x = ↑n |
CategoryTheory.Functor.IsStronglyCocartesian.map_isHomLift | Mathlib.CategoryTheory.FiberedCategory.Cocartesian | ∀ {𝒮 : Type u₁} {𝒳 : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} 𝒮] [inst_1 : CategoryTheory.Category.{v₂, u₂} 𝒳]
(p : CategoryTheory.Functor 𝒳 𝒮) {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) [inst_2 : p.IsStronglyCocartesian f φ]
{S' : 𝒮} {b' : 𝒳} {g : S ⟶ S'} {f' : R ⟶ S'} (hf' : f' = CategoryTheory.CategoryStruct.comp f g) (φ' : a ⟶ b')
[inst_3 : p.IsHomLift f' φ'], p.IsHomLift g (CategoryTheory.Functor.IsStronglyCocartesian.map p f φ hf' φ') |
Mathlib.Meta.Finset.ProveEmptyOrConsResult.empty.elim | Mathlib.Tactic.NormNum.BigOperators | {u : Lean.Level} →
{α : Q(Type u)} →
{s : Q(Finset «$α»)} →
{motive : Mathlib.Meta.Finset.ProveEmptyOrConsResult s → Sort u} →
(t : Mathlib.Meta.Finset.ProveEmptyOrConsResult s) →
t.ctorIdx = 0 → ((pf : Q(«$s» = ∅)) → motive (Mathlib.Meta.Finset.ProveEmptyOrConsResult.empty pf)) → motive t |
_private.Mathlib.LinearAlgebra.ExteriorAlgebra.Basic.0.ExteriorAlgebra.map_injective.match_1_1 | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | ∀ {R : Type u_3} [inst : CommRing R] {M : Type u_1} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type u_2}
[inst_3 : AddCommGroup N] [inst_4 : Module R N] {f : M →ₗ[R] N} (motive : (∃ g, g ∘ₗ f = LinearMap.id) → Prop)
(hf : ∃ g, g ∘ₗ f = LinearMap.id), (∀ (w : N →ₗ[R] M) (hgf : w ∘ₗ f = LinearMap.id), motive ⋯) → motive hf |
_private.Lean.Meta.Coe.0.Lean.Meta.coerceSimple?.match_1 | Lean.Meta.Coe | (motive : Lean.LOption (Lean.Expr × List Lean.Name) → Sort u_1) →
(__do_lift : Lean.LOption (Lean.Expr × List Lean.Name)) →
((result : Lean.Expr) → (snd : List Lean.Name) → motive (Lean.LOption.some (result, snd))) →
(Unit → motive Lean.LOption.none) → (Unit → motive Lean.LOption.undef) → motive __do_lift |
AddActionSemiHomClass.mk | Mathlib.GroupTheory.GroupAction.Hom | ∀ {F : Type u_8} {M : outParam (Type u_9)} {N : outParam (Type u_10)} {φ : outParam (M → N)} {X : outParam (Type u_11)}
{Y : outParam (Type u_12)} [inst : VAdd M X] [inst_1 : VAdd N Y] [inst_2 : FunLike F X Y],
(∀ (f : F) (c : M) (x : X), f (c +ᵥ x) = φ c +ᵥ f x) → AddActionSemiHomClass F φ X Y |
Filter.smul.instNeBot | Mathlib.Order.Filter.Pointwise | ∀ {α : Type u_2} {β : Type u_3} [inst : SMul α β] {f : Filter α} {g : Filter β} [f.NeBot] [g.NeBot], (f • g).NeBot |
Asymptotics.SuperpolynomialDecay.add | Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | ∀ {α : Type u_1} {β : Type u_2} {l : Filter α} {k f g : α → β} [inst : TopologicalSpace β] [inst_1 : CommSemiring β]
[ContinuousAdd β],
Asymptotics.SuperpolynomialDecay l k f →
Asymptotics.SuperpolynomialDecay l k g → Asymptotics.SuperpolynomialDecay l k (f + g) |
Subgroup.quotientiInfSubgroupOfEmbedding._proof_1 | Mathlib.GroupTheory.Coset.Basic | ∀ {α : Type u_1} [inst : Group α] {ι : Type u_2} (f : ι → Subgroup α) (i : ι), iInf f ≤ f i |
EMetric.ball_disjoint | Mathlib.Topology.EMetricSpace.Defs | ∀ {α : Type u} [inst : PseudoEMetricSpace α] {x y : α} {ε₁ ε₂ : ENNReal},
ε₁ + ε₂ ≤ edist x y → Disjoint (Metric.eball x ε₁) (Metric.eball y ε₂) |
_private.Init.Data.Order.Lemmas.0.Std.instAssociativeMaxOfIsLinearOrderOfLawfulOrderMax._simp_1 | Init.Data.Order.Lemmas | ∀ {α : Type u} [inst : Max α] [inst_1 : LE α] [Std.LawfulOrderSup α] {a b c : α}, (a ⊔ b ≤ c) = (a ≤ c ∧ b ≤ c) |
le_mul_of_one_le_left' | Mathlib.Algebra.Order.Monoid.Unbundled.Basic | ∀ {α : Type u_1} [inst : MulOneClass α] [inst_1 : LE α] [MulRightMono α] {a b : α}, 1 ≤ b → a ≤ b * a |
Std.LawfulEqOrd.compare_eq_iff_eq._simp_1 | Init.Data.Order.Ord | ∀ {α : Type u} [inst : Ord α] [Std.LawfulEqOrd α] {a b : α}, (compare a b = Ordering.eq) = (a = b) |
Std.HashSet.Raw.WF.casesOn | Std.Data.HashSet.Raw | {α : Type u} →
[inst : BEq α] →
[inst_1 : Hashable α] →
{m : Std.HashSet.Raw α} → {motive : m.WF → Sort u_1} → (t : m.WF) → ((out : m.inner.WF) → motive ⋯) → motive t |
_private.Mathlib.Tactic.Simproc.ExistsAndEq.0.ExistsAndEq.withNestedExistsElim.match_1 | Mathlib.Tactic.Simproc.ExistsAndEq | (motive : List ExistsAndEq.VarQ → Sort u_1) →
(exs : List ExistsAndEq.VarQ) →
(Unit → motive []) →
((u : Lean.Level) → (β : Q(Sort u)) → (b : Q(«$β»)) → (tl : List ExistsAndEq.VarQ) → motive (⟨u, ⟨β, b⟩⟩ :: tl)) →
motive exs |
Configuration.HasPoints.lineCount_eq_pointCount | Mathlib.Combinatorics.Configuration | ∀ {P : Type u_1} {L : Type u_2} [inst : Membership P L] [Configuration.HasPoints P L] [inst_2 : Fintype P]
[inst_3 : Fintype L],
Fintype.card P = Fintype.card L →
∀ {p : P} {l : L}, p ∉ l → Configuration.lineCount L p = Configuration.pointCount P l |
Set.FiniteExhaustion._sizeOf_inst | Mathlib.Data.Set.FiniteExhaustion | {α : Type u_1} → (s : Set α) → [SizeOf α] → [(a : α) → SizeOf (s a)] → SizeOf s.FiniteExhaustion |
IntermediateField.induction_on_adjoin | Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | ∀ {F : Type u_1} [inst : Field F] {E : Type u_2} [inst_1 : Field E] [inst_2 : Algebra F E] [FiniteDimensional F E]
(P : IntermediateField F E → Prop),
P ⊥ →
(∀ (K : IntermediateField F E) (x : E), P K → P (IntermediateField.restrictScalars F (↥K)⟮x⟯)) →
∀ (K : IntermediateField F E), P K |
Lean.Sym.UInt16.lt_eq_true | Init.Sym.Lemmas | ∀ (a b : UInt16), decide (a < b) = true → (a < b) = True |
AddMonoidAlgebra.mapRangeRingHom._proof_1 | Mathlib.Algebra.MonoidAlgebra.MapDomain | ∀ {R : Type u_3} {S : Type u_1} (M : Type u_2) [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : AddMonoid M]
(f : R →+* S), Finsupp.mapRange ⇑f ⋯ 1 = 1 |
Std.HashMap.Raw.getKey_eq | Std.Data.HashMap.RawLemmas | ∀ {α : Type u} {β : Type v} {m : Std.HashMap.Raw α β} [inst : BEq α] [inst_1 : Hashable α] [LawfulBEq α],
m.WF → ∀ {k : α} (h' : k ∈ m), m.getKey k h' = k |
AlgebraicGeometry.Scheme.basicOpen_restrict | Mathlib.AlgebraicGeometry.Scheme | ∀ (X : AlgebraicGeometry.Scheme) {V U : X.Opens} (i : V ⟶ U) (f : ↑(X.presheaf.obj (Opposite.op U))),
X.basicOpen (TopCat.Presheaf.restrict f i) ≤ X.basicOpen f |
CategoryTheory.Pretriangulated.instHasFiniteCoproducts | Mathlib.CategoryTheory.Triangulated.Pretriangulated | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C]
[inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C]
[inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C],
CategoryTheory.Limits.HasFiniteCoproducts C |
Std.DTreeMap.Const.getKey?_insertMany_list_of_mem | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {cmp : α → α → Ordering} {β : Type v} {t : Std.DTreeMap α (fun x => β) cmp} [Std.TransCmp cmp]
{l : List (α × β)} {k k' : 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.Const.insertMany t l).getKey? k' = some k |
ProbabilityTheory.complexMGF_id_gaussianReal | Mathlib.Probability.Distributions.Gaussian.Real | ∀ {μ : ℝ} {v : NNReal} (z : ℂ),
ProbabilityTheory.complexMGF id (ProbabilityTheory.gaussianReal μ v) z = Complex.exp (z * ↑μ + ↑↑v * z ^ 2 / 2) |
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le | Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | ∀ {ι : Type u_1} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E} {a : ℝ} {c : E}
{l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
[CompleteSpace E] [intervalIntegral.FTCFilter a l l'],
StronglyMeasurableAtFilter f l' μ →
Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (nhds c) →
Filter.Tendsto u lt l →
Filter.Tendsto v lt l →
u ≤ᶠ[lt] v →
(fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - μ.real (Set.Ioc (u t) (v t)) • c) =o[lt] fun t =>
μ.real (Set.Ioc (u t) (v t)) |
Matroid.freeOn.eq_1 | Mathlib.Combinatorics.Matroid.Constructions | ∀ {α : Type u_1} (E : Set α), Matroid.freeOn E = (Matroid.loopyOn E)✶ |
Lean.Elab.Tactic.Do.SpecAttr.SpecTheorems.isErased | Lean.Elab.Tactic.Do.Attr | Lean.Elab.Tactic.Do.SpecAttr.SpecTheorems → Lean.Elab.Tactic.Do.SpecAttr.SpecProof → Bool |
PSigma.Lex.preorder._proof_4 | Mathlib.Data.PSigma.Order | ∀ {ι : Type u_2} {α : ι → Type u_1} [inst : Preorder ι] [inst_1 : (i : ι) → Preorder (α i)] (a b : Σₗ' (i : ι), α i),
a < b ↔ a ≤ b ∧ ¬b ≤ a |
«command#long_names_» | Mathlib.Util.LongNames | Lean.ParserDescr |
OrderedFinpartition.eraseMiddle._proof_11 | Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | ∀ {n : ℕ} (c : OrderedFinpartition (n + 1)) (hc : Set.range (c.emb 0) ≠ {0}) (i j : Fin c.length),
i < j →
(fun m =>
if h : m = c.index 0 then
(c.emb m
(Fin.cast ⋯ ⟨Function.update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) m - 1, ⋯⟩.succ)).pred
⋯
else
(c.emb m (Fin.cast ⋯ ⟨Function.update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) m - 1, ⋯⟩)).pred
⋯)
i <
(fun m =>
if h : m = c.index 0 then
(c.emb m
(Fin.cast ⋯ ⟨Function.update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) m - 1, ⋯⟩.succ)).pred
⋯
else
(c.emb m (Fin.cast ⋯ ⟨Function.update c.partSize (c.index 0) (c.partSize (c.index 0) - 1) m - 1, ⋯⟩)).pred
⋯)
j |
Sub.noConfusion | Init.Prelude | {P : Sort u_1} →
{α : Type u} → {t : Sub α} → {α' : Type u} → {t' : Sub α'} → α = α' → t ≍ t' → Sub.noConfusionType P t t' |
BooleanAlgebra.recOn | Mathlib.Order.BooleanAlgebra.Defs | {α : Type u} →
{motive : BooleanAlgebra α → Sort u_1} →
(t : BooleanAlgebra α) →
([toDistribLattice : DistribLattice α] →
[toCompl : Compl α] →
[toSDiff : SDiff α] →
[toHImp : HImp α] →
[toTop : Top α] →
[toBot : Bot α] →
(inf_compl_le_bot : ∀ (x : α), x ⊓ xᶜ ≤ ⊥) →
(top_le_sup_compl : ∀ (x : α), ⊤ ≤ x ⊔ xᶜ) →
(le_top : ∀ (a : α), a ≤ ⊤) →
(bot_le : ∀ (a : α), ⊥ ≤ a) →
(sdiff_eq : ∀ (x y : α), x \ y = x ⊓ yᶜ) →
(himp_eq : ∀ (x y : α), x ⇨ y = y ⊔ xᶜ) →
motive
{ toDistribLattice := toDistribLattice, toCompl := toCompl, toSDiff := toSDiff,
toHImp := toHImp, toTop := toTop, toBot := toBot,
inf_compl_le_bot := inf_compl_le_bot, top_le_sup_compl := top_le_sup_compl,
le_top := le_top, bot_le := bot_le, sdiff_eq := sdiff_eq, himp_eq := himp_eq }) →
motive t |
RingHom.liftOfRightInverse_comp | Mathlib.RingTheory.Ideal.Maps | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} [inst : Ring A] [inst_1 : Ring B] [inst_2 : Ring C] (f : A →+* B)
(f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }),
((f.liftOfRightInverse f_inv hf) g).comp f = ↑g |
LinearMap.extendScalarsOfIsLocalizationEquiv._proof_8 | Mathlib.RingTheory.Localization.Module | ∀ {R : Type u_3} [inst : CommSemiring R] (A : Type u_4) [inst_1 : CommSemiring A] [inst_2 : Algebra R A] {M : Type u_1}
{N : Type u_2} [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M] [IsScalarTower R A M]
[inst_7 : AddCommMonoid N] [inst_8 : Module R N] [inst_9 : Module A N] [IsScalarTower R A N],
LinearMap.CompatibleSMul M N R A |
Int.Linear.Poly.insert | Init.Data.Int.Linear | ℤ → Int.Linear.Var → Int.Linear.Poly → Int.Linear.Poly |
_private.Mathlib.Topology.Category.CompHaus.EffectiveEpi.0.CompHaus.effectiveEpiFamily_tfae.match_1_1 | Mathlib.Topology.Category.CompHaus.EffectiveEpi | ∀ {α : Type} [inst : Finite α] {B : CompHaus} (X : α → CompHaus) (π : (a : α) → X a ⟶ B)
(motive : CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc π) → Prop)
(x : CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc π)),
(∀ (x : CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc π)), motive x) → motive x |
_private.Mathlib.Combinatorics.Matroid.Constructions.0.Matroid.empty_isBase_iff._simp_1_2 | Mathlib.Combinatorics.Matroid.Constructions | ∀ {α : Type u_1} (M : Matroid α), M.3 ∅ = True |
IsContDiffImplicitAt.implicitFunctionData | Mathlib.Analysis.Calculus.ImplicitContDiff | {𝕜 : Type u_1} →
[inst : RCLike 𝕜] →
{E : Type u_2} →
[inst_1 : NormedAddCommGroup E] →
[inst_2 : NormedSpace 𝕜 E] →
[inst_3 : CompleteSpace E] →
{F : Type u_3} →
[inst_4 : NormedAddCommGroup F] →
[inst_5 : NormedSpace 𝕜 F] →
[inst_6 : CompleteSpace F] →
{G : Type u_4} →
[inst_7 : NormedAddCommGroup G] →
[inst_8 : NormedSpace 𝕜 G] →
[inst_9 : CompleteSpace G] →
{n : WithTop ℕ∞} →
{f : E × F → G} →
{f' : E × F →L[𝕜] G} →
{a : E × F} → IsContDiffImplicitAt n f f' a → ImplicitFunctionData 𝕜 (E × F) E G |
_private.Mathlib.AlgebraicTopology.DoldKan.Degeneracies.0.AlgebraicTopology.DoldKan.σ_comp_P_eq_zero._simp_1_5 | Mathlib.AlgebraicTopology.DoldKan.Degeneracies | ∀ {α : Type u_1} [inst : Fintype α] (x : α), (x ∈ Finset.univ) = True |
Tactic.NormNum.int_lcm_helper | Mathlib.Tactic.NormNum.GCD | ∀ {x y : ℤ} {x' y' d : ℕ}, x.natAbs = x' → y.natAbs = y' → x'.lcm y' = d → x.lcm y = d |
IsZGroup.isCyclic_commutator | Mathlib.GroupTheory.SpecificGroups.ZGroup | ∀ (G : Type u_1) [inst : Group G] [Finite G] [IsZGroup G], IsCyclic ↥(commutator G) |
CategoryTheory.Limits.limit.isLimit | Mathlib.CategoryTheory.Limits.HasLimits | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
{C : Type u} →
[inst_1 : CategoryTheory.Category.{v, u} C] →
(F : CategoryTheory.Functor J C) →
[inst_2 : CategoryTheory.Limits.HasLimit F] →
CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.limit.cone F) |
LinearMap.compMultilinearMap_add | Mathlib.LinearAlgebra.Multilinear.Basic | ∀ {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [inst : Semiring R]
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃]
[inst_4 : (i : ι) → Module R (M₁ i)] [inst_5 : Module R M₂] [inst_6 : Module R M₃] (g : M₂ →ₗ[R] M₃)
(f₁ f₂ : MultilinearMap R M₁ M₂), g.compMultilinearMap (f₁ + f₂) = g.compMultilinearMap f₁ + g.compMultilinearMap f₂ |
_private.Mathlib.Order.Interval.Set.Basic.0.Set.instNoMinOrderElemIio.match_1 | Mathlib.Order.Interval.Set.Basic | ∀ {α : Type u_1} [inst : Preorder α] {a : α} (a_1 : ↑(Set.Iio a)) (motive : (∃ b, b < ↑a_1) → Prop) (x : ∃ b, b < ↑a_1),
(∀ (b : α) (hb : b < ↑a_1), motive ⋯) → motive x |
BitVec.ushiftRight_and_distrib | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} (x y : BitVec w) (n : ℕ), (x &&& y) >>> n = x >>> n &&& y >>> n |
ENNReal.toReal_eq_toReal_iff' | Mathlib.Data.ENNReal.Basic | ∀ {x y : ENNReal}, x ≠ ⊤ → y ≠ ⊤ → (x.toReal = y.toReal ↔ x = y) |
List.zipWithM.loop | Init.Data.List.Control | {m : Type u → Type v} →
[Monad m] → {α : Type w} → {β : Type x} → {γ : Type u} → (α → β → m γ) → List α → List β → Array γ → m (List γ) |
_private.Mathlib.MeasureTheory.Integral.CircleIntegral.0.circleIntegrable_sub_inv_iff._simp_1_1 | Mathlib.MeasureTheory.Integral.CircleIntegral | ∀ {G : Type u_1} [inst : DivInvMonoid G] (x : G), x⁻¹ = x ^ (-1) |
GradedRing.projZeroRingHom'_surjective | Mathlib.RingTheory.GradedAlgebra.Basic | ∀ {ι : Type u_1} {A : Type u_3} {σ : Type u_4} [inst : Semiring A] [inst_1 : DecidableEq ι] [inst_2 : AddCommMonoid ι]
[inst_3 : PartialOrder ι] [inst_4 : CanonicallyOrderedAdd ι] [inst_5 : SetLike σ A] [inst_6 : AddSubmonoidClass σ A]
(𝒜 : ι → σ) [inst_7 : GradedRing 𝒜], Function.Surjective ⇑(GradedRing.projZeroRingHom' 𝒜) |
factorPowSucc.isUnit_of_isUnit_image | Mathlib.RingTheory.Ideal.Quotient.PowTransition | ∀ {R : Type u_3} [inst : CommRing R] {I : Ideal R} {n : ℕ},
n > 0 → ∀ {a : R ⧸ I ^ (n + 1)}, IsUnit ((Ideal.Quotient.factorPow I ⋯) a) → IsUnit a |
Filter.tendstoIxxClass_principal | Mathlib.Order.Filter.Interval | ∀ {α : Type u_1} {s t : Set α} {Ixx : α → α → Set α},
Filter.TendstoIxxClass Ixx (Filter.principal s) (Filter.principal t) ↔ ∀ x ∈ s, ∀ y ∈ s, Ixx x y ⊆ t |
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