name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
_private.Init.Data.Array.InsertionSort.0.Array.insertionSort.swapLoop.match_1 | Init.Data.Array.InsertionSort | (motive : ℕ → Sort u_1) → (j : ℕ) → (j = 0 → motive 0) → ((j' : ℕ) → j = j'.succ → motive j'.succ) → motive j |
RingHomId.eq_id | Mathlib.Algebra.Ring.CompTypeclasses | ∀ {R : Type u_4} {inst : Semiring R} {σ : R →+* R} [self : RingHomId σ], σ = RingHom.id R |
LinearEquiv.piRing | Mathlib.LinearAlgebra.Pi | (R : Type u) →
(M : Type v) →
(ι : Type x) →
[inst : Semiring R] →
(S : Type u_4) →
[Fintype ι] →
[DecidableEq ι] →
[inst_3 : Semiring S] →
[inst_4 : AddCommMonoid M] →
[inst_5 : Module R M] →
[inst_6 : Module S M] → [inst_7 : SMulCommClass R S M] → ((ι → R) →ₗ[R] M) ≃ₗ[S] ι → M |
Std.Internal.List.maxKeyD_le_maxKeyD_insertEntry | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
{l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
l.isEmpty = false →
∀ {k : α} {v : β k} {fallback : α},
(compare (Std.Internal.List.maxKeyD l fallback)
(Std.Internal.List.maxKeyD (Std.Internal.List.insertEntry k v l) fallback)).isLE =
true |
Finset.prod_ite_of_false | Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | ∀ {ι : Type u_1} {M : Type u_3} {s : Finset ι} [inst : CommMonoid M] {p : ι → Prop} [inst_1 : DecidablePred p],
(∀ x ∈ s, ¬p x) → ∀ (f g : ι → M), (∏ x ∈ s, if p x then f x else g x) = ∏ x ∈ s, g x |
_private.Lean.Meta.ExprTraverse.0.Lean.Meta.traverseForallWithPos.visit.match_1 | Lean.Meta.ExprTraverse | (motive : Lean.Expr → Sort u_1) →
(x : Lean.Expr) →
((n : Lean.Name) → (d b : Lean.Expr) → (c : Lean.BinderInfo) → motive (Lean.Expr.forallE n d b c)) →
((e : Lean.Expr) → motive e) → motive x |
Equiv.recOn | Mathlib.Logic.Equiv.Defs | {α : Sort u_1} →
{β : Sort u_2} →
{motive : α ≃ β → Sort u} →
(t : α ≃ β) →
((toFun : α → β) →
(invFun : β → α) →
(left_inv : Function.LeftInverse invFun toFun) →
(right_inv : Function.RightInverse invFun toFun) →
motive { toFun := toFun, invFun := invFun, left_inv := left_inv, right_inv := right_inv }) →
motive t |
Continuous.cfcₙ_nnreal'._auto_3 | Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | Lean.Syntax |
topologicalAddGroup_inf | Mathlib.Topology.Algebra.Group.Basic | ∀ {G : Type w} [inst : AddGroup G] {t₁ t₂ : TopologicalSpace G},
IsTopologicalAddGroup G → IsTopologicalAddGroup G → IsTopologicalAddGroup G |
Acc.casesOn | Init.WF | {α : Sort u} →
{r : α → α → Prop} →
{motive : (a : α) → Acc r a → Sort u_1} →
{a : α} → (t : Acc r a) → ((x : α) → (h : ∀ (y : α), r y x → Acc r y) → motive x ⋯) → motive a t |
Function.Embedding.instAddAction._proof_1 | Mathlib.GroupTheory.GroupAction.Embedding | ∀ {α : Type u_1} {β : Type u_2}, Function.Injective fun f => ⇑f |
Multiset.powerset._proof_1 | Mathlib.Data.Multiset.Powerset | ∀ {α : Type u_1} (x x_1 : List α),
(List.isSetoid α) x x_1 →
Quot.mk (⇑(List.isSetoid (Multiset α))) (Multiset.powersetAux x) =
Quot.mk (⇑(List.isSetoid (Multiset α))) (Multiset.powersetAux x_1) |
ZFSet.Insert.match_5 | Mathlib.SetTheory.ZFC.Basic | ∀ (α : Type u_1) (A : α → PSet.{u_1}) (α_1 : Type u_1) (A_1 : α_1 → PSet.{u_1}) (b : (PSet.mk α_1 A_1).Type)
(motive : (∃ a, (A a).Equiv (A_1 b)) → Prop) (x : ∃ a, (A a).Equiv (A_1 b)),
(∀ (a : α) (ha : (A a).Equiv (A_1 b)), motive ⋯) → motive x |
List.Sublist.flatMap | Mathlib.Data.List.Flatten | ∀ {α : Type u_1} {β : Type u_2} {l₁ l₂ : List α},
l₁.Sublist l₂ → ∀ (f : α → List β), (List.flatMap f l₁).Sublist (List.flatMap f l₂) |
Ring.zsmul | Mathlib.Algebra.Ring.Defs | {R : Type u} → [self : Ring R] → ℤ → R → R |
MeasureTheory.StronglyAdapted.progMeasurable_of_continuous | Mathlib.Probability.Process.Adapted | ∀ {Ω : Type u_1} {ι : Type u_2} {m : MeasurableSpace Ω} [inst : Preorder ι] {f : MeasureTheory.Filtration ι m}
{β : Type u_3} [inst_1 : TopologicalSpace β] {u : ι → Ω → β} [inst_2 : TopologicalSpace ι]
[TopologicalSpace.MetrizableSpace ι] [SecondCountableTopology ι] [inst_5 : MeasurableSpace ι] [OpensMeasurableSpace ι]
[TopologicalSpace.PseudoMetrizableSpace β],
MeasureTheory.StronglyAdapted f u → (∀ (ω : Ω), Continuous fun i => u i ω) → MeasureTheory.ProgMeasurable f u |
Equiv.simpleGraph | Mathlib.Combinatorics.SimpleGraph.Maps | {V : Type u_1} → {W : Type u_2} → V ≃ W → SimpleGraph V ≃ SimpleGraph W |
Std.Internal.List.minKey!_insertEntryIfNew_le_minKey! | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : Ord α] [Std.TransOrd α] [inst_2 : BEq α] [Std.LawfulBEqOrd α]
[inst_4 : Inhabited α] {l : List ((a : α) × β a)},
Std.Internal.List.DistinctKeys l →
l.isEmpty = false →
∀ {k : α} {v : β k},
(compare (Std.Internal.List.minKey! (Std.Internal.List.insertEntryIfNew k v l))
(Std.Internal.List.minKey! l)).isLE =
true |
QuadraticAlgebra.instNonUnitalNonAssocSemiring | Mathlib.Algebra.QuadraticAlgebra.Defs | {R : Type u_1} → {a b : R} → [NonUnitalNonAssocSemiring R] → NonUnitalNonAssocSemiring (QuadraticAlgebra R a b) |
CompactlySupportedContinuousMap.smulc_apply | Mathlib.Topology.ContinuousMap.CompactlySupported | ∀ {α : Type u_2} {β : Type u_3} {γ : Type u_4} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β]
[inst_2 : Zero β] [inst_3 : TopologicalSpace γ] [inst_4 : SMulZeroClass γ β] [inst_5 : ContinuousSMul γ β]
{F : Type u_5} [inst_6 : FunLike F α γ] [inst_7 : ContinuousMapClass F α γ] (f : F)
(g : CompactlySupportedContinuousMap α β) (x : α), (f • g) x = f x • g x |
Lean.Meta.Grind.SplitInfo.arg | Lean.Meta.Tactic.Grind.Types | Lean.Expr → Lean.Expr → ℕ → Lean.Expr → Lean.Meta.Grind.SplitSource → Lean.Meta.Grind.SplitInfo |
_private.Mathlib.Data.PFun.0.PFun.mem_prodLift._simp_1_6 | Mathlib.Data.PFun | ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, p x ∧ b) = ((∃ x, p x) ∧ b) |
AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeftIsLimit | Mathlib.Geometry.RingedSpace.OpenImmersion | {X Y Z : AlgebraicGeometry.LocallyRingedSpace} →
(f : X ⟶ Z) →
(g : Y ⟶ Z) →
[H : AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion f] →
CategoryTheory.Limits.IsLimit (AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeft f g) |
List.dropWhile.eq_def | Init.Data.List.TakeDrop | ∀ {α : Type u} (p : α → Bool) (x : List α),
List.dropWhile p x =
match x with
| [] => []
| a :: l =>
match p a with
| true => List.dropWhile p l
| false => a :: l |
Finsupp.mem_submodule_iff | Mathlib.LinearAlgebra.Finsupp.Pi | ∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_5}
(S : α → Submodule R M) (x : α →₀ M), x ∈ Finsupp.submodule S ↔ ∀ (i : α), x i ∈ S i |
Submonoid.val_mem_of_mem_units | Mathlib.Algebra.Group.Submonoid.Units | ∀ {M : Type u_1} [inst : Monoid M] (S : Submonoid M) {x : Mˣ}, x ∈ S.units → ↑x ∈ S |
Finsupp.mem_neLocus | Mathlib.Data.Finsupp.NeLocus | ∀ {α : Type u_1} {N : Type u_3} [inst : DecidableEq α] [inst_1 : DecidableEq N] [inst_2 : Zero N] {f g : α →₀ N}
{a : α}, a ∈ f.neLocus g ↔ f a ≠ g a |
Std.DTreeMap.isSome_minKey?_of_mem | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp] {k : α},
k ∈ t → t.minKey?.isSome = true |
CategoryTheory.Functor.mapTriangleCommShiftIso_inv_app_hom₁ | Mathlib.CategoryTheory.Triangulated.Functor | ∀ {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] [inst_2 : CategoryTheory.HasShift C ℤ]
[inst_3 : CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [inst_4 : F.CommShift ℤ]
[inst_5 : CategoryTheory.Preadditive C] [inst_6 : CategoryTheory.Preadditive D] [inst_7 : F.Additive] (n : ℤ)
(X : CategoryTheory.Pretriangulated.Triangle C),
((F.mapTriangleCommShiftIso n).inv.app X).hom₁ = (CategoryTheory.Functor.commShiftIso F n).inv.app X.obj₁ |
Nat.Partrec.Code.ofNatCode.eq_4 | Mathlib.Computability.PartrecCode | Nat.Partrec.Code.ofNatCode 3 = Nat.Partrec.Code.right |
_private.Init.Data.Int.DivMod.Lemmas.0.Int.fdiv_fmod_unique'._proof_1_1 | Init.Data.Int.DivMod.Lemmas | ∀ {b : ℤ}, b < 0 → ¬0 < -b → False |
Lean.Doc.Syntax.directive._regBuiltin.Lean.Doc.Syntax.directive.docString_1 | Lean.DocString.Syntax | IO Unit |
ArchimedeanClass.mk_nonneg_of_le_of_le_of_archimedean | Mathlib.Algebra.Order.Ring.Archimedean | ∀ {R : Type u_1} [inst : LinearOrder R] [inst_1 : CommRing R] [inst_2 : IsStrictOrderedRing R] {S : Type u_3}
[inst_3 : LinearOrder S] [inst_4 : CommRing S] [IsStrictOrderedRing S] [Archimedean S] (f : S →+*o R) {x : R}
{r s : S}, f r ≤ x → x ≤ f s → 0 ≤ ArchimedeanClass.mk x |
CommRingCat.Colimits.instCommRingColimitType._proof_9 | Mathlib.Algebra.Category.Ring.Colimits | ∀ {J : Type u_1} [inst : CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CommRingCat)
(x : CommRingCat.Colimits.ColimitType F), 0 * x = 0 |
Quaternion.imJ_star | Mathlib.Algebra.Quaternion | ∀ {R : Type u_3} [inst : CommRing R] (a : Quaternion R), (star a).imJ = -a.imJ |
List.splitAtD.go._sunfold | Batteries.Data.List.Basic | {α : Type u_1} → α → ℕ → List α → List α → List α × List α |
Lean.Meta.LazyDiscrTree.recOn | Lean.Meta.LazyDiscrTree | {α : Type} →
{motive : Lean.Meta.LazyDiscrTree α → Sort u} →
(t : Lean.Meta.LazyDiscrTree α) →
((tries : Array (Lean.Meta.LazyDiscrTree.Trie α)) →
(roots : Std.HashMap Lean.Meta.LazyDiscrTree.Key Lean.Meta.LazyDiscrTree.TrieIndex) →
motive { tries := tries, roots := roots }) →
motive t |
Mathlib.Tactic.Ring.ringCleanupRef | Mathlib.Tactic.Ring.Basic | IO.Ref (Lean.Expr → Lean.MetaM Lean.Expr) |
VitaliFamily.FineSubfamilyOn.index | Mathlib.MeasureTheory.Covering.VitaliFamily | {X : Type u_1} →
[inst : PseudoMetricSpace X] →
{m0 : MeasurableSpace X} →
{μ : MeasureTheory.Measure X} →
{v : VitaliFamily μ} → {f : X → Set (Set X)} → {s : Set X} → v.FineSubfamilyOn f s → Set (X × Set X) |
SimpleGraph.Walk.IsHamiltonian.fintype._proof_1 | Mathlib.Combinatorics.SimpleGraph.Hamiltonian | ∀ {α : Type u_1} [inst : DecidableEq α] {G : SimpleGraph α} {a b : α} {p : G.Walk a b},
p.IsHamiltonian → ∀ (x : α), x ∈ p.support.toFinset |
_private.Lean.Elab.DocString.Builtin.0.Lean.Doc.suggestName.match_4 | Lean.Elab.DocString.Builtin | (motive : Lean.Exception → Sort u_1) → (ex : Lean.Exception) → ((x : Lean.Exception) → motive x) → motive ex |
Nat.odd_sub._simp_1 | Mathlib.Algebra.Ring.Parity | ∀ {m n : ℕ}, n ≤ m → Odd (m - n) = (Odd m ↔ Even n) |
CategoryTheory.Functor.elementsFunctor_map | Mathlib.CategoryTheory.Elements | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : CategoryTheory.Functor C (Type w)} (n : X ⟶ Y),
CategoryTheory.Functor.elementsFunctor.map n = (CategoryTheory.NatTrans.mapElements n).toCatHom |
WithZeroMulInt.toNNReal_le_one_iff | Mathlib.Data.Int.WithZero | ∀ {e : NNReal} {m : WithZero (Multiplicative ℤ)} (he : 1 < e), (WithZeroMulInt.toNNReal ⋯) m ≤ 1 ↔ m ≤ 1 |
Algebra.transcendental_ringHom_iff_of_comp_eq | Mathlib.RingTheory.Algebraic.Basic | ∀ {R : Type u} {S : Type u_1} {A : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Ring A]
[inst_3 : Algebra R A] {B : Type u_2} [inst_4 : Ring B] [inst_5 : Algebra S B] {FRS : Type u_3} {FAB : Type u_4}
[inst_6 : EquivLike FRS R S] [inst_7 : RingEquivClass FRS R S] [inst_8 : EquivLike FAB A B]
[inst_9 : RingEquivClass FAB A B] (f : FRS) (g : FAB),
(algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A) → (Algebra.Transcendental S B ↔ Algebra.Transcendental R A) |
padicValRat.of_int | Mathlib.NumberTheory.Padics.PadicVal.Basic | ∀ {p : ℕ} {z : ℤ}, padicValRat p ↑z = ↑(padicValInt p z) |
orderBornology_isBounded._simp_1 | Mathlib.Topology.Order.Bornology | ∀ {α : Type u_1} {s : Set α} [inst : Lattice α] [inst_1 : Nonempty α], Bornology.IsBounded s = (BddBelow s ∧ BddAbove s) |
Std.Tactic.BVDecide.LRAT.Internal.Formula.rupAdd_sound | Std.Tactic.BVDecide.LRAT.Internal.Formula.Class | ∀ {α : outParam (Type u)} {β : outParam (Type v)} {inst : Std.Tactic.BVDecide.LRAT.Internal.Clause α β} {σ : Type w}
{inst_1 : Std.Tactic.BVDecide.LRAT.Internal.Entails α σ} [self : Std.Tactic.BVDecide.LRAT.Internal.Formula α β σ]
(f : σ) (c : β) (rupHints : Array ℕ) (f' : σ),
Std.Tactic.BVDecide.LRAT.Internal.Formula.ReadyForRupAdd f →
Std.Tactic.BVDecide.LRAT.Internal.Formula.performRupAdd f c rupHints = (f', true) →
Std.Tactic.BVDecide.LRAT.Internal.Liff α f f' |
CategoryTheory.Precoherent.recOn | Mathlib.CategoryTheory.Sites.Coherent.Basic | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{motive : CategoryTheory.Precoherent C → Sort u} →
(t : CategoryTheory.Precoherent C) →
((pullback :
∀ {B₁ B₂ : C} (f : B₂ ⟶ B₁) (α : Type) [Finite α] (X₁ : α → C) (π₁ : (a : α) → X₁ a ⟶ B₁),
CategoryTheory.EffectiveEpiFamily X₁ π₁ →
∃ β,
∃ (_ : Finite β),
∃ X₂ π₂,
CategoryTheory.EffectiveEpiFamily X₂ π₂ ∧
∃ i ι,
∀ (b : β),
CategoryTheory.CategoryStruct.comp (ι b) (π₁ (i b)) =
CategoryTheory.CategoryStruct.comp (π₂ b) f) →
motive ⋯) →
motive t |
max_mul_mul_left | Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | ∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : Mul α] [MulLeftMono α] (a b c : α), max (a * b) (a * c) = a * max b c |
ProbabilityTheory.Kernel.ae_compProd_iff | Mathlib.Probability.Kernel.Composition.CompProd | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ]
{η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {p : β × γ → Prop},
MeasurableSet {x | p x} →
((∀ᵐ (bc : β × γ) ∂(κ.compProd η) a, p bc) ↔ ∀ᵐ (b : β) ∂κ a, ∀ᵐ (c : γ) ∂η (a, b), p (b, c)) |
Equiv.forall_congr' | Mathlib.Logic.Equiv.Defs | ∀ {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (e : α ≃ β),
(∀ (b : β), p (e.symm b) ↔ q b) → ((∀ (a : α), p a) ↔ ∀ (b : β), q b) |
CategoryTheory.congr_app | Mathlib.CategoryTheory.NatTrans | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{F G : CategoryTheory.Functor C D} {α β : CategoryTheory.NatTrans F G}, α = β → ∀ (X : C), α.app X = β.app X |
Fintype.linearIndependent_iffₛ | Mathlib.LinearAlgebra.LinearIndependent.Defs | ∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M] [inst_3 : Fintype ι],
LinearIndependent R v ↔ ∀ (f g : ι → R), ∑ i, f i • v i = ∑ i, g i • v i → ∀ (i : ι), f i = g i |
Group.nilpotencyClass_of_not_nilpotent | Mathlib.GroupTheory.Nilpotent | ∀ {G : Type u_1} [inst : Group G], ¬Group.IsNilpotent G → Group.nilpotencyClass G = 0 |
CategoryTheory.Functor.PreservesLeftKanExtension.mk._flat_ctor | Mathlib.CategoryTheory.Functor.KanExtension.Preserves | ∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} A]
[inst_1 : CategoryTheory.Category.{v_2, u_2} B] [inst_2 : CategoryTheory.Category.{v_3, u_3} C]
[inst_3 : CategoryTheory.Category.{v_4, u_4} D] {G : CategoryTheory.Functor B D} {F : CategoryTheory.Functor A B}
{L : CategoryTheory.Functor A C},
(∀ (F' : CategoryTheory.Functor C B) (α : F ⟶ L.comp F') [F'.IsLeftKanExtension α],
(F'.comp G).IsLeftKanExtension
(CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight α G) (L.associator F' G).hom)) →
G.PreservesLeftKanExtension F L |
_private.Mathlib.Data.List.Cycle.0.List.next_eq_getElem._proof_1_7 | Mathlib.Data.List.Cycle | ∀ {α : Type u_1} [inst : DecidableEq α] {l : List α} {a : α},
a ∈ l →
∀ (hl : l ≠ []),
¬(List.idxOf a l + 1) % l.length + 1 ≤ l.dropLast.length →
(List.idxOf a l + 1) % l.length - l.dropLast.length < [l.getLast ⋯].length |
_private.Mathlib.Algebra.IsPrimePow.0.not_isPrimePow_zero._simp_1_4 | Mathlib.Algebra.IsPrimePow | ∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c) |
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap_assoc | Mathlib.AlgebraicGeometry.ColimitsOver | ∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [inst : P.IsStableUnderBaseChange]
[inst_1 : P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1}
[inst_2 : CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover}
[inst_3 : CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [inst_4 : AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰]
(d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (a : J) {Z : P.Over ⊤ (𝒰.X j)}
(h : (d.cocone j).pt ⟶ Z),
CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).map ((d.cocone i).ι.app a))
(CategoryTheory.CategoryStruct.comp (d.transitionMap hij) h) =
CategoryTheory.CategoryStruct.comp ((d.trans hij).app a)
(CategoryTheory.CategoryStruct.comp ((d.cocone j).ι.app a) h) |
Real.expPartialHomeomorph_target | Mathlib.Analysis.SpecialFunctions.Log.Basic | Real.expPartialHomeomorph.target = Set.Ioi 0 |
IsCompl.compl_eq_iff | Mathlib.Order.BooleanAlgebra.Basic | ∀ {α : Type u} {x y z : α} [inst : BooleanAlgebra α], IsCompl x y → (zᶜ = y ↔ z = x) |
Array.all_iff_forall | Init.Data.Array.Lemmas | ∀ {α : Type u_1} {p : α → Bool} {as : Array α} {start stop : ℕ},
as.all p start stop = true ↔ ∀ (i : ℕ) (x : i < as.size), start ≤ i ∧ i < stop → p as[i] = true |
AddAction.sigmaFixedByEquivOrbitsProdAddGroup._proof_1 | Mathlib.GroupTheory.GroupAction.Quotient | ∀ (α : Type u_1) (β : Type u_2) [inst : AddGroup α] [inst_1 : AddAction α β] (x : α × β),
x.1 +ᵥ x.2 = x.2 ↔ x.1 +ᵥ x.2 = x.2 |
_private.Mathlib.Data.Rat.Sqrt.0.Rat.exists_mul_self.match_1_1 | Mathlib.Data.Rat.Sqrt | ∀ (x : ℚ) (motive : (∃ q, q * q = x) → Prop) (x_1 : ∃ q, q * q = x), (∀ (n : ℚ) (hn : n * n = x), motive ⋯) → motive x_1 |
Mathlib.Tactic._aux_Mathlib_Tactic_Core___macroRules_Mathlib_Tactic_tacticRepeat1__1 | Mathlib.Tactic.Core | Lean.Macro |
AlgebraicGeometry.instAddCommGroupObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInType | Mathlib.AlgebraicGeometry.StructureSheaf | {R M : Type u} →
[inst : CommRing R] →
[inst_1 : AddCommGroup M] →
[inst_2 : Module R M] →
(U : (TopologicalSpace.Opens ↑(AlgebraicGeometry.PrimeSpectrum.Top R))ᵒᵖ) →
AddCommGroup ((AlgebraicGeometry.structureSheafInType R M).obj.obj U) |
HahnSeries.toOrderTopSubOnePos | Mathlib.RingTheory.HahnSeries.Summable | {Γ : Type u_1} →
{R : Type u_3} →
[inst : AddCommMonoid Γ] →
[inst_1 : LinearOrder Γ] →
[inst_2 : IsOrderedCancelAddMonoid Γ] →
[inst_3 : CommRing R] → {x : HahnSeries Γ R} → 0 < (x - 1).orderTop → ↥(HahnSeries.orderTopSubOnePos Γ R) |
infEDist_inv | Mathlib.Analysis.Normed.Group.Pointwise | ∀ {E : Type u_1} [inst : SeminormedCommGroup E] (x : E) (s : Set E), Metric.infEDist x⁻¹ s = Metric.infEDist x s⁻¹ |
instBornologyPUnit._proof_1 | Mathlib.Topology.Bornology.Basic | ⊥ ≤ Filter.cofinite |
Lean.SerialMessage.ctorIdx | Lean.Message | Lean.SerialMessage → ℕ |
Char.lt | Init.Data.Char.Basic | Char → Char → Prop |
Lean.Grind.CommRing.Stepwise.div_cert.eq_1 | Init.Grind.Ring.CommSolver | ∀ (p₁ : Lean.Grind.CommRing.Poly) (k : ℤ) (p : Lean.Grind.CommRing.Poly),
Lean.Grind.CommRing.Stepwise.div_cert p₁ k p = (!k.beq' 0).and' ((Lean.Grind.CommRing.Poly.mulConst_k k p).beq' p₁) |
_private.Lean.Shell.0.Lean.displayHelp | Lean.Shell | Bool → IO Unit |
_private.Mathlib.FieldTheory.AlgebraicClosure.0.le_algebraicClosure_iff._simp_1_1 | Mathlib.FieldTheory.AlgebraicClosure | ∀ {F : Type u_1} {E : Type u_2} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {x : E},
(x ∈ algebraicClosure F E) = IsAlgebraic F x |
_private.Mathlib.Topology.CWComplex.Classical.Basic.0.Topology.RelCWComplex.disjoint_base_iUnion_openCell._simp_1_2 | Mathlib.Topology.CWComplex.Classical.Basic | ∀ {α : Type u_1} {ι : Sort u_5} {s : ι → Set α}, (⋃ i, s i = ∅) = ∀ (i : ι), s i = ∅ |
Array.forIn' | Init.Data.Array.Basic | {α : Type u} →
{β : Type v} → {m : Type v → Type w} → [Monad m] → (as : Array α) → β → ((a : α) → a ∈ as → β → m (ForInStep β)) → m β |
Std.ExtDTreeMap.Const.getEntryLT._proof_1 | Std.Data.ExtDTreeMap.Basic | ∀ {α : Type u_1} {cmp : α → α → Ordering} {β : Type u_2} [inst : Std.TransCmp cmp]
(t : Std.ExtDTreeMap α (fun x => β) cmp) (k : α),
(∃ a ∈ t, cmp a k = Ordering.lt) →
∀ (m : Std.DTreeMap α (fun x => β) cmp),
t = Std.ExtDTreeMap.mk m → ∃ a ∈ Std.ExtDTreeMap.mk m, cmp a k = Ordering.lt |
Real.logb_neg_of_base_lt_one | Mathlib.Analysis.SpecialFunctions.Log.Base | ∀ {b x : ℝ}, 0 < b → b < 1 → 1 < x → Real.logb b x < 0 |
CategoryTheory.equivEssImageOfReflective_inverse | Mathlib.CategoryTheory.Adjunction.Reflective | ∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
{i : CategoryTheory.Functor D C} [inst_2 : CategoryTheory.Reflective i],
CategoryTheory.equivEssImageOfReflective.inverse = i.essImage.ι.comp (CategoryTheory.reflector i) |
AddEquiv.mk.sizeOf_spec | Mathlib.Algebra.Group.Equiv.Defs | ∀ {A : Type u_9} {B : Type u_10} [inst : Add A] [inst_1 : Add B] [inst_2 : SizeOf A] [inst_3 : SizeOf B]
(toEquiv : A ≃ B) (map_add' : ∀ (x y : A), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y),
sizeOf { toEquiv := toEquiv, map_add' := map_add' } = 1 + sizeOf toEquiv |
HomotopicalAlgebra.LeftHomotopyRel.postcomp | Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C}
[inst_1 : HomotopicalAlgebra.CategoryWithWeakEquivalences C] {f g : X ⟶ Y},
HomotopicalAlgebra.LeftHomotopyRel f g →
∀ {Z : C} (p : Y ⟶ Z),
HomotopicalAlgebra.LeftHomotopyRel (CategoryTheory.CategoryStruct.comp f p)
(CategoryTheory.CategoryStruct.comp g p) |
_private.Init.Data.String.Basic.0.String.Pos.toSlice_le._simp_1_1 | Init.Data.String.Basic | ∀ {s : String.Slice} {l r : s.Pos}, (l ≤ r) = (l.offset ≤ r.offset) |
HomologicalComplex₂.D₁_totalShift₂XIso_hom | Mathlib.Algebra.Homology.TotalComplexShift | ∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
(K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (y : ℤ) [inst_2 : K.HasTotal (ComplexShape.up ℤ)]
(n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁'),
CategoryTheory.CategoryStruct.comp (((HomologicalComplex₂.shiftFunctor₂ C y).obj K).D₁ (ComplexShape.up ℤ) n₀ n₁)
(K.totalShift₂XIso y n₁ n₁' h₁).hom =
y.negOnePow •
CategoryTheory.CategoryStruct.comp (K.totalShift₂XIso y n₀ n₀' h₀).hom (K.D₁ (ComplexShape.up ℤ) n₀' n₁') |
ByteArray.extract_eq_empty_iff | Init.Data.ByteArray.Lemmas | ∀ {b : ByteArray} {i j : ℕ}, b.extract i j = ByteArray.empty ↔ min j b.size ≤ i |
_private.Mathlib.RingTheory.Ideal.GoingUp.0.Ideal.IsIntegralClosure.comap_ne_bot.match_1_1 | Mathlib.RingTheory.Ideal.GoingUp | ∀ {A : Type u_1} [inst : CommRing A] {I : Ideal A} (motive : (∃ x ∈ I, x ≠ 0) → Prop) (x : ∃ x ∈ I, x ≠ 0),
(∀ (x : A) (x_mem : x ∈ I) (x_ne_zero : x ≠ 0), motive ⋯) → motive x |
Lean.instInhabitedAuxParentProjectionInfo.default | Lean.ProjFns | Lean.AuxParentProjectionInfo |
_private.Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension.0.IsOpen.exists_contDiff_support_eq._simp_1_1 | Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | ∀ {α : Type u} {a : α} {s : Set α}, ({a} ⊆ s) = (a ∈ s) |
_private.Aesop.Forward.State.0.Aesop.instBEqRawHyp.beq.match_1 | Aesop.Forward.State | (motive : Aesop.RawHyp → Aesop.RawHyp → Sort u_1) →
(x x_1 : Aesop.RawHyp) →
((a b : Lean.FVarId) → motive (Aesop.RawHyp.fvarId a) (Aesop.RawHyp.fvarId b)) →
((a b : Aesop.Substitution) → motive (Aesop.RawHyp.patSubst a) (Aesop.RawHyp.patSubst b)) →
((x x_2 : Aesop.RawHyp) → motive x x_2) → motive x x_1 |
CategoryTheory.Monad.id._proof_1 | Mathlib.CategoryTheory.Monad.Basic | ∀ (C : Type u_2) [inst : CategoryTheory.Category.{u_1, u_2} C] (X : C),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Functor.id C).map ((CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.id C)).app X))
((CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.id C)).app X) =
CategoryTheory.CategoryStruct.comp
((CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.id C)).app ((CategoryTheory.Functor.id C).obj X))
((CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.id C)).app X) |
ProbabilityTheory.IndepFun.map_mul_eq_map_mconv_map₀ | Mathlib.Probability.Independence.Basic | ∀ {Ω : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {M : Type u_10} [inst : Monoid M]
[inst_1 : MeasurableSpace M] [MeasurableMul₂ M] [MeasureTheory.IsFiniteMeasure μ] {f g : Ω → M},
AEMeasurable f μ →
AEMeasurable g μ →
ProbabilityTheory.IndepFun f g μ →
MeasureTheory.Measure.map (f * g) μ = (MeasureTheory.Measure.map f μ).mconv (MeasureTheory.Measure.map g μ) |
multiplicity_addValuation_apply | Mathlib.RingTheory.Valuation.PrimeMultiplicity | ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] {p : R} {hp : Prime p} {r : R},
(multiplicity_addValuation hp) r = emultiplicity p r |
ContDiffAt.exists_forall_mem_closedBall_exists_eq_forall_mem_Ioo_hasDerivAt | Mathlib.Analysis.ODE.PicardLindelof | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {f : E → E} {x₀ : E},
ContDiffAt ℝ 1 f x₀ →
∀ (t₀ : ℝ),
∃ r > 0,
∃ ε > 0, ∀ x ∈ Metric.closedBall x₀ r, ∃ α, α t₀ = x ∧ ∀ t ∈ Set.Ioo (t₀ - ε) (t₀ + ε), HasDerivAt α (f (α t)) t |
Path.Homotopy.transAssoc._proof_4 | Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | ⟨Path.Homotopy.transAssocReparamAux 1, Path.Homotopy.transAssoc._proof_3⟩ = 1 |
Std.Tactic.BVDecide.Normalize.BitVec.beq_one_eq_ite' | Std.Tactic.BVDecide.Normalize.Bool | ∀ {b : Bool} {a : BitVec 1}, (b == (a == 1#1)) = (a == bif b then 1#1 else 0#1) |
HasFibers.instFaithfulFibι | Mathlib.CategoryTheory.FiberedCategory.HasFibers | ∀ {𝒮 : Type u₁} {𝒳 : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} 𝒮] [inst_1 : CategoryTheory.Category.{v₂, u₂} 𝒳]
(p : CategoryTheory.Functor 𝒳 𝒮) [inst_2 : HasFibers p] (S : 𝒮), (HasFibers.ι S).Faithful |
CategoryTheory.InjectiveResolution.toRightDerivedZero'._proof_2 | Mathlib.CategoryTheory.Abelian.RightDerived | ∀ {C : Type u_4} [inst : CategoryTheory.Category.{u_3, u_4} C] {D : Type u_2}
[inst_1 : CategoryTheory.Category.{u_1, u_2} D] [inst_2 : CategoryTheory.Abelian C]
[inst_3 : CategoryTheory.Abelian D] {X : C} (P : CategoryTheory.InjectiveResolution X)
(F : CategoryTheory.Functor C D) [inst_4 : F.Additive],
CategoryTheory.CategoryStruct.comp (F.map (P.ι.f 0))
(((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex).d 0 1) =
0 |
CategoryTheory.ParametrizedAdjunction.rec | Mathlib.CategoryTheory.Adjunction.Parametrized | {C₁ : Type u₁} →
{C₂ : Type u₂} →
{C₃ : Type u₃} →
[inst : CategoryTheory.Category.{v₁, u₁} C₁] →
[inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] →
[inst_2 : CategoryTheory.Category.{v₃, u₃} C₃] →
{F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₃)} →
{G : CategoryTheory.Functor C₁ᵒᵖ (CategoryTheory.Functor C₃ C₂)} →
{motive : (F ⊣₂ G) → Sort u} →
((adj : (X₁ : C₁) → F.obj X₁ ⊣ G.obj (Opposite.op X₁)) →
(unit_whiskerRight_map :
∀ {X₁ Y₁ : C₁} (f : X₁ ⟶ Y₁),
CategoryTheory.CategoryStruct.comp (adj X₁).unit
(CategoryTheory.Functor.whiskerRight (F.map f) (G.obj (Opposite.op X₁))) =
CategoryTheory.CategoryStruct.comp (adj Y₁).unit ((F.obj Y₁).whiskerLeft (G.map f.op))) →
motive { adj := adj, unit_whiskerRight_map := unit_whiskerRight_map }) →
(t : F ⊣₂ G) → motive t |
Turing.PartrecToTM2.tr.eq_2 | Mathlib.Computability.TuringMachine.ToPartrec | ∀ (k : Turing.PartrecToTM2.K') (f : Option Turing.PartrecToTM2.Γ' → Option Turing.PartrecToTM2.Γ')
(q : Turing.PartrecToTM2.Λ'),
Turing.PartrecToTM2.tr (Turing.PartrecToTM2.Λ'.push k f q) =
Turing.TM2.Stmt.branch (fun s => (f s).isSome)
(Turing.TM2.Stmt.push k (fun s => (f s).getD default) (Turing.TM2.Stmt.goto fun x => q))
(Turing.TM2.Stmt.goto fun x => q) |
Std.DTreeMap.getKeyD_minKey! | Std.Data.DTreeMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {t : Std.DTreeMap α β cmp} [Std.TransCmp cmp]
[inst : Inhabited α], t.isEmpty = false → ∀ {fallback : α}, t.getKeyD t.minKey! fallback = t.minKey! |
PMF.seq.eq_1 | Mathlib.Probability.ProbabilityMassFunction.Constructions | ∀ {α : Type u_1} {β : Type u_2} (q : PMF (α → β)) (p : PMF α), q.seq p = q.bind fun m => p.bind fun a => PMF.pure (m a) |
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