name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
_private.Mathlib.RingTheory.Polynomial.Resultant.Basic.0.Polynomial.resultant_add_mul_monomial_right._proof_1_25 | Mathlib.RingTheory.Polynomial.Resultant.Basic | ∀ (m n k : ℕ) (j₁ : Fin (m + n)) (j₂ : Fin m),
(↑j₂ ≤ ↑j₁ ∧ ↑j₁ ≤ ↑j₂ + n) ∧ k ≤ ↑j₁ - ↑j₂ → ¬(↑j₂ + k ≤ ↑j₁ ∧ ↑j₁ ≤ ↑j₂ + (k + m)) → m < ↑j₁ - ↑j₂ - k |
Mathlib.Tactic.Translate.TranslationInfo.recOn | Mathlib.Tactic.Translate.Core | {motive : Mathlib.Tactic.Translate.TranslationInfo → Sort u} →
(t : Mathlib.Tactic.Translate.TranslationInfo) →
((translation : Lean.Name) →
(reorder : Mathlib.Tactic.Translate.Reorder) →
(relevantArg : Mathlib.Tactic.Translate.RelevantArg) →
motive { translation := translation, reorder := reorder, relevantArg := relevantArg }) →
motive t |
MeasurableSet.mem | Mathlib.MeasureTheory.MeasurableSpace.Constructions | ∀ {α : Type u_1} {s : Set α} [inst : MeasurableSpace α], MeasurableSet s → Measurable fun x => x ∈ s |
dist_pi_const_le | Mathlib.Topology.MetricSpace.Pseudo.Pi | ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoMetricSpace α] [inst_1 : Fintype β] (a b : α),
(dist (fun x => a) fun x => b) ≤ dist a b |
_private.Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension.0.IsOpen.exists_contDiff_support_eq._simp_1_15 | Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | ∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],
(∀ᶠ (x : α) in Filter.atTop, p x) = ∃ a, ∀ b ≥ a, p b |
_private.Mathlib.Topology.Algebra.Polynomial.0.Polynomial.coeff_le_of_roots_le._simp_1_1 | Mathlib.Topology.Algebra.Polynomial | ∀ {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 |
_private.Mathlib.Analysis.InnerProductSpace.Defs.0.InnerProductSpace.Core.«_aux_Mathlib_Analysis_InnerProductSpace_Defs___macroRules__private_Mathlib_Analysis_InnerProductSpace_Defs_0_InnerProductSpace_Core_term_†_1» | Mathlib.Analysis.InnerProductSpace.Defs | Lean.Macro |
_private.Mathlib.Computability.TMToPartrec.0.Turing.PartrecToTM2.trStmts₁.match_1.eq_4 | Mathlib.Computability.TMToPartrec | ∀ (motive : Turing.PartrecToTM2.Λ' → Sort u_1) (p : Turing.PartrecToTM2.Γ' → Bool) (k : Turing.PartrecToTM2.K')
(q : Turing.PartrecToTM2.Λ')
(h_1 :
(Q : Turing.PartrecToTM2.Λ') →
(p : Turing.PartrecToTM2.Γ' → Bool) →
(k₁ k₂ : Turing.PartrecToTM2.K') →
(q : Turing.PartrecToTM2.Λ') →
Q = Turing.PartrecToTM2.Λ'.move p k₁ k₂ q → motive (Turing.PartrecToTM2.Λ'.move p k₁ k₂ q))
(h_2 :
(Q : Turing.PartrecToTM2.Λ') →
(k : Turing.PartrecToTM2.K') →
(s : Option Turing.PartrecToTM2.Γ' → Option Turing.PartrecToTM2.Γ') →
(q : Turing.PartrecToTM2.Λ') →
Q = Turing.PartrecToTM2.Λ'.push k s q → motive (Turing.PartrecToTM2.Λ'.push k s q))
(h_3 :
(Q : Turing.PartrecToTM2.Λ') →
(q : Option Turing.PartrecToTM2.Γ' → Turing.PartrecToTM2.Λ') →
Q = Turing.PartrecToTM2.Λ'.read q → motive (Turing.PartrecToTM2.Λ'.read q))
(h_4 :
(Q : Turing.PartrecToTM2.Λ') →
(p : Turing.PartrecToTM2.Γ' → Bool) →
(k : Turing.PartrecToTM2.K') →
(q : Turing.PartrecToTM2.Λ') →
Q = Turing.PartrecToTM2.Λ'.clear p k q → motive (Turing.PartrecToTM2.Λ'.clear p k q))
(h_5 : (Q q : Turing.PartrecToTM2.Λ') → Q = q.copy → motive q.copy)
(h_6 : (Q q : Turing.PartrecToTM2.Λ') → Q = q.succ → motive q.succ)
(h_7 : (Q q₁ q₂ : Turing.PartrecToTM2.Λ') → Q = q₁.pred q₂ → motive (q₁.pred q₂))
(h_8 :
(Q : Turing.PartrecToTM2.Λ') →
(k : Turing.PartrecToTM2.Cont') → Q = Turing.PartrecToTM2.Λ'.ret k → motive (Turing.PartrecToTM2.Λ'.ret k)),
(match Turing.PartrecToTM2.Λ'.clear p k q with
| Q@h:(Turing.PartrecToTM2.Λ'.move p k₁ k₂ q) => h_1 Q p k₁ k₂ q h
| Q@h:(Turing.PartrecToTM2.Λ'.push k s q) => h_2 Q k s q h
| Q@h:(Turing.PartrecToTM2.Λ'.read q) => h_3 Q q h
| Q@h:(Turing.PartrecToTM2.Λ'.clear p k q) => h_4 Q p k q h
| Q@h:q.copy => h_5 Q q h
| Q@h:q.succ => h_6 Q q h
| Q@h:(q₁.pred q₂) => h_7 Q q₁ q₂ h
| Q@h:(Turing.PartrecToTM2.Λ'.ret k) => h_8 Q k h) =
h_4 (Turing.PartrecToTM2.Λ'.clear p k q) p k q ⋯ |
Real.sign_eq_zero_iff._simp_1 | Mathlib.Data.Real.Sign | ∀ {r : ℝ}, (r.sign = 0) = (r = 0) |
minimal_nonempty_open_eq_singleton | Mathlib.Topology.Separation.Basic | ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
IsOpen s → s.Nonempty → (∀ t ⊆ s, t.Nonempty → IsOpen t → t = s) → ∃ x, s = {x} |
left_iff_dite_iff | Init.PropLemmas | ∀ {p : Prop} [inst : Decidable p] {x : Prop} {y : ¬p → Prop}, (x ↔ if h : p then x else y h) ↔ ∀ (h : ¬p), x ↔ y h |
GradedMonoid.list_prod_map_eq_dProd | Mathlib.Algebra.GradedMonoid | ∀ {ι : Type u_1} {α : Type u_2} {A : ι → Type u_3} [inst : AddMonoid ι] [inst_1 : GradedMonoid.GMonoid A] (l : List α)
(f : α → GradedMonoid A),
(List.map f l).prod =
GradedMonoid.mk (l.dProdIndex fun i => (f i).fst) (l.dProd (fun i => (f i).fst) fun i => (f i).snd) |
CategoryTheory.IsPullback.of_hasPullback | Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs | ∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[inst_1 : CategoryTheory.Limits.HasPullback f g],
CategoryTheory.IsPullback (CategoryTheory.Limits.pullback.fst f g) (CategoryTheory.Limits.pullback.snd f g) f g |
ProbabilityTheory.Kernel.IndepFun.comp | Mathlib.Probability.Independence.Kernel.IndepFun | ∀ {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {γ : Type u_6} {γ' : Type u_7} {mα : MeasurableSpace α}
{mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'}
{mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {mγ : MeasurableSpace γ} {mγ' : MeasurableSpace γ'} {φ : β → γ}
{ψ : β' → γ'},
ProbabilityTheory.Kernel.IndepFun f g κ μ →
Measurable φ → Measurable ψ → ProbabilityTheory.Kernel.IndepFun (φ ∘ f) (ψ ∘ g) κ μ |
_private.Std.Data.Internal.List.Associative.0.Std.Internal.List.getKey_eraseKey._simp_1_1 | Std.Data.Internal.List.Associative | ∀ {α : Type u} {β : α → Type v} [inst : BEq α] {l : List ((a : α) × β a)} {a : α}
(h : Std.Internal.List.containsKey a l = true), some (Std.Internal.List.getKey a l h) = Std.Internal.List.getKey? a l |
CategoryTheory.Under.pushout_obj | Mathlib.CategoryTheory.Comma.Over.Pullback | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
[inst_1 : CategoryTheory.Limits.HasPushoutsAlong f] (x : CategoryTheory.Under X),
(CategoryTheory.Under.pushout f).obj x = CategoryTheory.Under.mk (CategoryTheory.Limits.pushout.inr x.hom f) |
Order.height_toDual | Mathlib.Order.KrullDimension | ∀ {α : Type u_1} [inst : Preorder α] (x : α), Order.height (OrderDual.toDual x) = Order.coheight x |
AddLocalization.ind | Mathlib.GroupTheory.MonoidLocalization.Basic | ∀ {M : Type u_1} [inst : AddCommMonoid M] {S : AddSubmonoid M} {p : AddLocalization S → Prop},
(∀ (y : M × ↥S), p (AddLocalization.mk y.1 y.2)) → ∀ (x : AddLocalization S), p x |
ProbabilityTheory.Kernel.partialTraj_comp_partialTraj | Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | ∀ {X : ℕ → Type u_1} {mX : (n : ℕ) → MeasurableSpace (X n)} {a b c : ℕ}
{κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))},
a ≤ b →
b ≤ c →
(ProbabilityTheory.Kernel.partialTraj κ b c).comp (ProbabilityTheory.Kernel.partialTraj κ a b) =
ProbabilityTheory.Kernel.partialTraj κ a c |
Ordinal.bsup_le | Mathlib.SetTheory.Ordinal.Family | ∀ {o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v}} {a : Ordinal.{max u v}},
(∀ (i : Ordinal.{u}) (h : i < o), f i h ≤ a) → o.bsup f ≤ a |
Lean.Elab.TacticInfo.goalsBefore | Lean.Elab.InfoTree.Types | Lean.Elab.TacticInfo → List Lean.MVarId |
RingOfIntegers.exponent | Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {K : Type u_1} → [inst : Field K] → NumberField.RingOfIntegers K → ℕ |
ContMDiff.piecewise | Mathlib.Geometry.Manifold.ContMDiff.Basic | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
[inst_4 : TopologicalSpace M] {E' : Type u_5} [inst_5 : NormedAddCommGroup E'] [inst_6 : NormedSpace 𝕜 E']
{H' : Type u_6} [inst_7 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7}
[inst_8 : TopologicalSpace M'] [inst_9 : ChartedSpace H M] [inst_10 : ChartedSpace H' M'] {n : WithTop ℕ∞}
{f g : M → M'} {s : Set M} [inst_11 : DecidablePred fun x => x ∈ s],
ContMDiff I I' n f → ContMDiff I I' n g → (∀ x ∈ frontier s, f =ᶠ[nhds x] g) → ContMDiff I I' n (s.piecewise f g) |
FirstOrder.Language.mk.injEq | Mathlib.ModelTheory.Basic | ∀ (Functions : ℕ → Type u) (Relations : ℕ → Type v) (Functions_1 : ℕ → Type u) (Relations_1 : ℕ → Type v),
({ Functions := Functions, Relations := Relations } = { Functions := Functions_1, Relations := Relations_1 }) =
(Functions = Functions_1 ∧ Relations = Relations_1) |
Mathlib.Tactic.Linarith.LinarithConfig.splitHypotheses._default | Mathlib.Tactic.Linarith.Frontend | Bool |
LLVM.Visibility._sizeOf_1 | Lean.Compiler.IR.LLVMBindings | LLVM.Visibility → ℕ |
NonUnitalCommRing.toNonUnitalCommSemiring._proof_2 | Mathlib.Algebra.Ring.Defs | ∀ {α : Type u_1} [s : NonUnitalCommRing α] (a b c : α), a * (b + c) = a * b + a * c |
Lean.removeRoot | Lean.Data.OpenDecl | Lean.Name → Lean.Name |
_private.Std.Data.DHashMap.Internal.AssocList.Lemmas.0.Std.DHashMap.Internal.AssocList.Const.alter.match_1.eq_1 | Std.Data.DHashMap.Internal.AssocList.Lemmas | ∀ {β : Type u_1} (motive : Option β → Sort u_2) (h_1 : Unit → motive none) (h_2 : (b : β) → motive (some b)),
(match none with
| none => h_1 ()
| some b => h_2 b) =
h_1 () |
_private.Mathlib.Algebra.Homology.ExactSequenceFour.0.CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac._proof_15 | Mathlib.Algebra.Homology.ExactSequenceFour | ∀ {n : ℕ} (k : ℕ),
autoParam (k ≤ n) CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac._auto_1 → k + 1 + 2 < n + 3 + 1 |
CategoryTheory.inclusion | Mathlib.CategoryTheory.ConnectedComponents | {J : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} J] →
(j : CategoryTheory.ConnectedComponents J) → CategoryTheory.Functor j.Component (CategoryTheory.Decomposed J) |
ContDiffMapSupportedInClass.casesOn | Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {B : Type u_5} →
{E : Type u_6} →
{F : Type u_7} →
[inst : NormedAddCommGroup E] →
[inst_1 : NormedAddCommGroup F] →
[inst_2 : NormedSpace ℝ E] →
[inst_3 : NormedSpace ℝ F] →
{n : ℕ∞} →
{K : TopologicalSpace.Compacts E} →
{motive : ContDiffMapSupportedInClass B E F n K → Sort u} →
(t : ContDiffMapSupportedInClass B E F n K) →
([toDFunLike : DFunLike B E fun x => F] →
(map_contDiff : ∀ (f : B), ContDiff ℝ ↑n ⇑f) →
(map_zero_on_compl : ∀ (f : B), Set.EqOn (⇑f) 0 (↑K)ᶜ) →
motive
{ toDFunLike := toDFunLike, map_contDiff := map_contDiff,
map_zero_on_compl := map_zero_on_compl }) →
motive t |
Lean.Meta.Cache._sizeOf_1 | Lean.Meta.Basic | Lean.Meta.Cache → ℕ |
ExteriorAlgebra.lift_symm_apply | Mathlib.LinearAlgebra.ExteriorAlgebra.Basic | ∀ (R : Type u1) [inst : CommRing R] {M : Type u2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {A : Type u_1}
[inst_3 : Semiring A] [inst_4 : Algebra R A] (a : ExteriorAlgebra R M →ₐ[R] A),
(ExteriorAlgebra.lift R).symm a = ⟨↑((CliffordAlgebra.lift 0).symm a), ⋯⟩ |
Mathlib.Tactic.IntervalCases.Methods.bisect._unsafe_rec | Mathlib.Tactic.IntervalCases | Mathlib.Tactic.IntervalCases.Methods →
Lean.MVarId →
Subarray Mathlib.Tactic.IntervalCases.IntervalCasesSubgoal →
Mathlib.Tactic.IntervalCases.Bound →
Mathlib.Tactic.IntervalCases.Bound → Lean.Expr → Lean.Expr → Lean.Expr → Lean.Expr → Lean.Expr → Lean.MetaM Unit |
_private.Mathlib.NumberTheory.FermatPsp.0.Nat.exists_infinite_pseudoprimes._proof_1_6 | Mathlib.NumberTheory.FermatPsp | ∀ (m : ℕ), 1 < 2 * (m + 2) |
Mathlib.Tactic.RingNF.RingMode.ctorElimType | Mathlib.Tactic.Ring.RingNF | {motive : Mathlib.Tactic.RingNF.RingMode → Sort u} → ℕ → Sort (max 1 u) |
Complex.lim_re | Mathlib.Analysis.Complex.Norm | ∀ (f : CauSeq ℂ fun x => ‖x‖), (Complex.cauSeqRe f).lim = f.lim.re |
enorm_prod_le_of_le | Mathlib.Analysis.Normed.Group.Basic | ∀ {ι : Type u_3} {ε : Type u_8} [inst : TopologicalSpace ε] [inst_1 : ESeminormedCommMonoid ε] (s : Finset ι)
{f : ι → ε} {n : ι → ENNReal}, (∀ b ∈ s, ‖f b‖ₑ ≤ n b) → ‖∏ b ∈ s, f b‖ₑ ≤ ∑ b ∈ s, n b |
_private.Mathlib.Analysis.Hofer.0._aux_Mathlib_Analysis_Hofer___macroRules__private_Mathlib_Analysis_Hofer_0_termD_1 | Mathlib.Analysis.Hofer | Lean.Macro |
_private.Mathlib.Data.Nat.Factorization.Basic.0.Nat.Ico_pow_dvd_eq_Ico_of_lt._simp_1_3 | Mathlib.Data.Nat.Factorization.Basic | ∀ {a c b : Prop}, (a ∧ c ↔ b ∧ c) = (c → (a ↔ b)) |
Lean.Elab.Term.ToDepElimPattern.State | Lean.Elab.Match | Type |
_private.Mathlib.Order.Directed.0.directedOn_iff_directed._simp_1_5 | Mathlib.Order.Directed | ∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b) |
MonoidHom.CompTriple.IsId.eq_id | Mathlib.Algebra.Group.Hom.CompTypeclasses | ∀ {M : Type u_1} {inst : Monoid M} {σ : M →* M} [self : MonoidHom.CompTriple.IsId σ], σ = MonoidHom.id M |
Equiv.prodPiEquivSumPi_apply | Mathlib.Logic.Equiv.Prod | ∀ {ι : Type u_9} {ι' : Type u_10} (π : ι → Type u) (π' : ι' → Type u)
(a : ((i : ι) → Sum.elim π π' (Sum.inl i)) × ((i' : ι') → Sum.elim π π' (Sum.inr i'))) (i : ι ⊕ ι'),
(Equiv.prodPiEquivSumPi π π') a i = (Equiv.sumPiEquivProdPi (Sum.elim π π')).symm a i |
CategoryTheory.Functor.descOfIsLeftKanExtension_fac_app | Mathlib.CategoryTheory.Functor.KanExtension.Basic | ∀ {C : Type u_1} {H : Type u_3} {D : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_3, u_3} H] [inst_2 : CategoryTheory.Category.{v_4, u_4} D]
(F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H}
(α : F ⟶ L.comp F') [inst_3 : F'.IsLeftKanExtension α] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C),
CategoryTheory.CategoryStruct.comp (α.app X) ((F'.descOfIsLeftKanExtension α G β).app (L.obj X)) = β.app X |
CategoryTheory.ShortComplex.LeftHomologyData.copy._proof_5 | Mathlib.Algebra.Homology.ShortComplex.LeftHomology | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S : CategoryTheory.ShortComplex C} (h : S.LeftHomologyData) {K' : C} (eK : K' ≅ h.K),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Limits.parallelPair (h.hi.lift (CategoryTheory.Limits.KernelFork.ofι S.f ⋯)) 0).map
CategoryTheory.Limits.WalkingParallelPairHom.left)
eK.symm.hom =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Iso.refl S.X₁).hom
((CategoryTheory.Limits.parallelPair
((CategoryTheory.Limits.IsKernel.isoKernel S.g (CategoryTheory.CategoryStruct.comp eK.hom h.i) h.hi eK
⋯).lift
(CategoryTheory.Limits.KernelFork.ofι S.f ⋯))
0).map
CategoryTheory.Limits.WalkingParallelPairHom.left) |
Lean.Grind.CommRing.Poly.insert.go.induct_unfolding | Init.Grind.Ring.CommSolver | ∀ (k : ℤ) (m : Lean.Grind.CommRing.Mon) (motive : Lean.Grind.CommRing.Poly → Lean.Grind.CommRing.Poly → Prop),
(∀ (k_1 : ℤ),
motive (Lean.Grind.CommRing.Poly.num k_1) (Lean.Grind.CommRing.Poly.add k m (Lean.Grind.CommRing.Poly.num k_1))) →
(∀ (k_1 : ℤ) (m_1 : Lean.Grind.CommRing.Mon) (p : Lean.Grind.CommRing.Poly),
m.grevlex m_1 = Ordering.eq →
have k := k + k_1;
(k == 0) = true → motive (Lean.Grind.CommRing.Poly.add k_1 m_1 p) p) →
(∀ (k_1 : ℤ) (m_1 : Lean.Grind.CommRing.Mon) (p : Lean.Grind.CommRing.Poly),
m.grevlex m_1 = Ordering.eq →
have k := k + k_1;
(k == 0) = false → motive (Lean.Grind.CommRing.Poly.add k_1 m_1 p) (Lean.Grind.CommRing.Poly.add k m p)) →
(∀ (k_1 : ℤ) (m_1 : Lean.Grind.CommRing.Mon) (p : Lean.Grind.CommRing.Poly),
m.grevlex m_1 = Ordering.gt →
motive (Lean.Grind.CommRing.Poly.add k_1 m_1 p)
(Lean.Grind.CommRing.Poly.add k m (Lean.Grind.CommRing.Poly.add k_1 m_1 p))) →
(∀ (k_1 : ℤ) (m_1 : Lean.Grind.CommRing.Mon) (p : Lean.Grind.CommRing.Poly),
m.grevlex m_1 = Ordering.lt →
motive p (Lean.Grind.CommRing.Poly.insert.go k m p) →
motive (Lean.Grind.CommRing.Poly.add k_1 m_1 p)
(Lean.Grind.CommRing.Poly.add k_1 m_1 (Lean.Grind.CommRing.Poly.insert.go k m p))) →
∀ (a : Lean.Grind.CommRing.Poly), motive a (Lean.Grind.CommRing.Poly.insert.go k m a) |
_private.Mathlib.LinearAlgebra.LinearPMap.0.LinearPMap.graph_map_fst_eq_domain._simp_1_6 | Mathlib.LinearAlgebra.LinearPMap | ∀ {α : Sort u_1} {p : α → Prop} {b : Prop}, (∃ x, p x ∧ b) = ((∃ x, p x) ∧ b) |
MeasureTheory.exp_neg_llr | Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | ∀ {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ]
[MeasureTheory.SigmaFinite ν],
μ.AbsolutelyContinuous ν → (fun x => Real.exp (-MeasureTheory.llr μ ν x)) =ᵐ[μ] fun x => (ν.rnDeriv μ x).toReal |
Lean.Compiler.LCNF.CtorFieldInfo.object.injEq | Lean.Compiler.LCNF.ToImpureType | ∀ (i : ℕ) (type : Lean.Expr) (i_1 : ℕ) (type_1 : Lean.Expr),
(Lean.Compiler.LCNF.CtorFieldInfo.object i type = Lean.Compiler.LCNF.CtorFieldInfo.object i_1 type_1) =
(i = i_1 ∧ type = type_1) |
ContDiff.fourierPowSMulRight | Mathlib.Analysis.Fourier.FourierTransformDeriv | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {V : Type u_2} {W : Type u_3}
[inst_2 : NormedAddCommGroup V] [inst_3 : NormedSpace ℝ V] [inst_4 : NormedAddCommGroup W] [inst_5 : NormedSpace ℝ W]
(L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} {k : WithTop ℕ∞},
ContDiff ℝ k f → ∀ (n : ℕ), ContDiff ℝ k fun v => VectorFourier.fourierPowSMulRight L f v n |
TopModuleCat | Mathlib.Algebra.Category.ModuleCat.Topology.Basic | (R : Type u) → [Ring R] → [TopologicalSpace R] → Type (max u (v + 1)) |
Lean.Lsp.Ipc.CallHierarchy.rec_2 | Lean.Data.Lsp.Ipc | {motive_1 : Lean.Lsp.Ipc.CallHierarchy → Sort u} →
{motive_2 : Array Lean.Lsp.Ipc.CallHierarchy → Sort u} →
{motive_3 : List Lean.Lsp.Ipc.CallHierarchy → Sort u} →
((item : Lean.Lsp.CallHierarchyItem) →
(fromRanges : Array Lean.Lsp.Range) →
(children : Array Lean.Lsp.Ipc.CallHierarchy) →
motive_2 children → motive_1 { item := item, fromRanges := fromRanges, children := children }) →
((toList : List Lean.Lsp.Ipc.CallHierarchy) → motive_3 toList → motive_2 { toList := toList }) →
motive_3 [] →
((head : Lean.Lsp.Ipc.CallHierarchy) →
(tail : List Lean.Lsp.Ipc.CallHierarchy) → motive_1 head → motive_3 tail → motive_3 (head :: tail)) →
(t : List Lean.Lsp.Ipc.CallHierarchy) → motive_3 t |
CategoryTheory.SmallObject.hasPushouts | Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (I : CategoryTheory.MorphismProperty C) (κ : Cardinal.{w})
[inst_1 : Fact κ.IsRegular] [inst_2 : OrderBot κ.ord.ToType] [I.IsCardinalForSmallObjectArgument κ],
CategoryTheory.Limits.HasPushouts C |
AddSubgroup.op.instNormal | Mathlib.Algebra.Group.Subgroup.MulOppositeLemmas | ∀ {G : Type u_2} [inst : AddGroup G] {H : AddSubgroup G} [H.Normal], H.op.Normal |
LinearMap.BilinForm.apply_apply_same_eq_zero_iff | Mathlib.LinearAlgebra.SesquilinearForm.Basic | ∀ {R : Type u_1} {M : Type u_5} [inst : CommRing R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R]
[inst_3 : AddCommGroup M] [inst_4 : Module R M] (B : LinearMap.BilinForm R M),
(∀ (x : M), 0 ≤ (B x) x) → LinearMap.IsSymm B → ∀ {x : M}, (B x) x = 0 ↔ x ∈ LinearMap.ker B |
AnalyticAt.comp_of_eq' | Mathlib.Analysis.Analytic.Composition | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
[inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
[inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {g : F → G} {f : E → F} {y : F} {x : E},
AnalyticAt 𝕜 g y → AnalyticAt 𝕜 f x → f x = y → AnalyticAt 𝕜 (fun z => g (f z)) x |
Equiv.piCongr'.eq_1 | Mathlib.Logic.Equiv.Basic | ∀ {α : Sort u_1} {β : Sort u_4} {W : α → Sort w} {Z : β → Sort z} (h₁ : α ≃ β) (h₂ : (b : β) → W (h₁.symm b) ≃ Z b),
h₁.piCongr' h₂ = (h₁.symm.piCongr fun b => (h₂ b).symm).symm |
Fin.cast_addNat | Init.Data.Fin.Lemmas | ∀ {n : ℕ} (m : ℕ) (i : Fin n), Fin.cast ⋯ (i.addNat m) = Fin.natAdd m i |
unitary.val_toUnits_apply | Mathlib.Algebra.Star.Unitary | ∀ {R : Type u_1} [inst : Monoid R] [inst_1 : StarMul R] (x : ↥(unitary R)), ↑(Unitary.toUnits x) = ↑x |
_private.Mathlib.Topology.Metrizable.Uniformity.0.UniformSpace.metrizable_uniformity._simp_1_5 | Mathlib.Topology.Metrizable.Uniformity | ∀ {α : Sort u_1} (a : α), (a = a) = True |
List.nodup_iff_forall_not_duplicate | Mathlib.Data.List.Duplicate | ∀ {α : Type u_1} {l : List α}, l.Nodup ↔ ∀ (x : α), ¬List.Duplicate x l |
LiouvilleWith.sub_nat_iff._simp_1 | Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | ∀ {p x : ℝ} {n : ℕ}, LiouvilleWith p (x - ↑n) = LiouvilleWith p x |
Pi.Lex.instCompleteLinearOrderLexForall._proof_19 | Mathlib.Order.CompleteLattice.PiLex | ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : LinearOrder ι] [inst_1 : (i : ι) → CompleteLinearOrder (α i)]
[inst_2 : WellFoundedLT ι] (a b : Lex ((i : ι) → α i)), compare a b = compareOfLessAndEq a b |
CategoryTheory.Subfunctor.Subpresheaf.toPresheaf_map_coe | Mathlib.CategoryTheory.Subfunctor.Basic | ∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor C (Type w)}
(G : CategoryTheory.Subfunctor F) (x x_1 : C) (i : x ⟶ x_1) (x_2 : ↑(G.obj x)),
↑(G.toFunctor.map i x_2) = F.map i ↑x_2 |
Std.TreeMap.Raw.insertMany_list_equiv_foldl | Std.Data.TreeMap.Raw.Lemmas | ∀ {α : Type u} {β : Type v} {cmp : α → α → Ordering} {t₁ : Std.TreeMap.Raw α β cmp} {l : List (α × β)},
(t₁.insertMany l).Equiv (List.foldl (fun acc p => acc.insert p.1 p.2) t₁ l) |
CategoryTheory.ObjectProperty.productTo | Mathlib.CategoryTheory.Generator.Basic | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
(P : CategoryTheory.ObjectProperty C) →
(X : C) → [inst_1 : CategoryTheory.Limits.HasProduct (P.productToFamily X)] → X ⟶ ∏ᶜ P.productToFamily X |
Finset.zero_mem_neg_add_iff._simp_1 | Mathlib.Algebra.Group.Pointwise.Finset.Basic | ∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : AddGroup α] {s t : Finset α}, (0 ∈ -t + s) = ¬Disjoint s t |
Nat.ppred | Mathlib.Data.Nat.PSub | ℕ → Option ℕ |
Matrix.instNonUnitalRing._proof_1 | Mathlib.Data.Matrix.Mul | ∀ {n : Type u_1} {α : Type u_2} [inst : NonUnitalRing α] (a b : Matrix n n α), a - b = a + -b |
Tree.noConfusionType | Mathlib.Data.Tree.Basic | Sort u_1 → {α : Type u} → Tree α → {α' : Type u} → Tree α' → Sort u_1 |
LinearEquiv._sizeOf_inst | Mathlib.Algebra.Module.Equiv.Defs | {R : Type u_14} →
{S : Type u_15} →
{inst : Semiring R} →
{inst_1 : Semiring S} →
(σ : R →+* S) →
{σ' : S →+* R} →
{inst_2 : RingHomInvPair σ σ'} →
{inst_3 : RingHomInvPair σ' σ} →
(M : Type u_16) →
(M₂ : Type u_17) →
{inst_4 : AddCommMonoid M} →
{inst_5 : AddCommMonoid M₂} →
{inst_6 : Module R M} →
{inst_7 : Module S M₂} →
[SizeOf R] → [SizeOf S] → [SizeOf M] → [SizeOf M₂] → SizeOf (M ≃ₛₗ[σ] M₂) |
Cycle.length_nil | Mathlib.Data.List.Cycle | ∀ {α : Type u_1}, Cycle.nil.length = 0 |
Lean.Elab.Command.elabEvalCoreUnsafe | Lean.Elab.BuiltinEvalCommand | Bool → Lean.Syntax → Lean.Syntax → Option Lean.Expr → Lean.Elab.Command.CommandElabM Unit |
CoxeterSystem.length_eq_one_iff | Mathlib.GroupTheory.Coxeter.Length | ∀ {B : Type u_1} {W : Type u_2} [inst : Group W] {M : CoxeterMatrix B} (cs : CoxeterSystem M W) {w : W},
cs.length w = 1 ↔ ∃ i, w = cs.simple i |
Multiplicative.mulAction_isPretransitive | Mathlib.Algebra.Group.Action.Pretransitive | ∀ {α : Type u_3} {β : Type u_4} [inst : AddMonoid α] [inst_1 : AddAction α β] [AddAction.IsPretransitive α β],
MulAction.IsPretransitive (Multiplicative α) β |
LawfulMonadAttach.eq_of_canReturn_pure | Init.Control.Lawful.MonadAttach.Lemmas | ∀ {m : Type u_1 → Type u_2} {α : Type u_1} [inst : Monad m] [inst_1 : MonadAttach m] [LawfulMonad m]
[LawfulMonadAttach m] {a b : α}, MonadAttach.CanReturn (pure a) b → a = b |
MeasureTheory.integral_prod_swap | Mathlib.MeasureTheory.Integral.Prod | ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β]
{μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [inst_2 : NormedAddCommGroup E] [MeasureTheory.SFinite ν]
[inst_4 : NormedSpace ℝ E] [MeasureTheory.SFinite μ] (f : α × β → E),
∫ (z : β × α), f z.swap ∂ν.prod μ = ∫ (z : α × β), f z ∂μ.prod ν |
Mathlib.Tactic.Linarith.Comp.scale | Mathlib.Tactic.Linarith.Datatypes | Mathlib.Tactic.Linarith.Comp → ℕ → Mathlib.Tactic.Linarith.Comp |
mul_le_of_mul_le_left | Mathlib.Algebra.Order.Monoid.Unbundled.Basic | ∀ {α : Type u_1} [inst : Mul α] [inst_1 : Preorder α] [MulLeftMono α] {a b c d : α}, a * b ≤ c → d ≤ b → a * d ≤ c |
_private.Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddSound.0.Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_fold_fn_preserves_induction_motive._simp_1_4 | Std.Tactic.BVDecide.LRAT.Internal.Formula.RupAddSound | ∀ {α : Type u_1} {β : Type u_2} {p : α × β → Prop}, (∀ (x : α × β), p x) = ∀ (a : α) (b : β), p (a, b) |
div_div_eq_mul_div | Mathlib.Algebra.Group.Basic | ∀ {α : Type u_1} [inst : DivisionMonoid α] (a b c : α), a / (b / c) = a * c / b |
Lean.Compiler.LCNF.AuxDeclCacheKey.casesOn | Lean.Compiler.LCNF.AuxDeclCache | {motive : Lean.Compiler.LCNF.AuxDeclCacheKey → Sort u} →
(t : Lean.Compiler.LCNF.AuxDeclCacheKey) →
((pu : Lean.Compiler.LCNF.Purity) → (decl : Lean.Compiler.LCNF.Decl pu) → motive { pu := pu, decl := decl }) →
motive t |
HomogeneousSubsemiring.ext | Mathlib.RingTheory.GradedAlgebra.Homogeneous.Subsemiring | ∀ {ι : Type u_1} {σ : Type u_2} {A : Type u_3} [inst : AddMonoid ι] [inst_1 : Semiring A] [inst_2 : SetLike σ A]
[inst_3 : AddSubmonoidClass σ A] {𝒜 : ι → σ} [inst_4 : DecidableEq ι] [inst_5 : GradedRing 𝒜]
{R S : HomogeneousSubsemiring 𝒜}, R.toSubsemiring = S.toSubsemiring → R = S |
CategoryTheory.Limits.IsLimit.pushoutOfHasExactLimitsOfShape._proof_2 | Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | ∀ {J : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} J] {C : Type u_4}
[inst_1 : CategoryTheory.Category.{u_3, u_4} C] [inst_2 : CategoryTheory.Limits.HasPushouts C]
[CategoryTheory.Limits.HasLimitsOfShape J C] {F : CategoryTheory.Functor J C} {c : CategoryTheory.Limits.Cone F}
{X : C} (f : c.pt ⟶ X),
CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.pushout c.π ((CategoryTheory.Functor.const J).map f)) |
perfectClosure.eq_bot_of_isSeparable | Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | ∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] [Algebra.IsSeparable F E],
perfectClosure F E = ⊥ |
_private.Mathlib.Combinatorics.SimpleGraph.Bipartite.0.SimpleGraph.IsBipartite.exists_isBipartiteWith._proof_1_3 | Mathlib.Combinatorics.SimpleGraph.Bipartite | NeZero (1 + 1) |
ContinuousLinearMap.flipMultilinearEquiv._proof_4 | Mathlib.Analysis.Normed.Module.Multilinear.Basic | ∀ (𝕜 : Type u_1) {ι : Type u_2} (E : ι → Type u_3) (G : Type u_5) (G' : Type u_4) [inst : NontriviallyNormedField 𝕜]
[inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)]
[inst_3 : SeminormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] [inst_5 : SeminormedAddCommGroup G']
[inst_6 : NormedSpace 𝕜 G'] [inst_7 : Fintype ι] (f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')),
‖(ContinuousLinearMap.flipMultilinearEquivₗ 𝕜 E G G').symm f‖ ≤ 1 * ‖f‖ |
Std.Internal.List.getEntry?._sunfold | Std.Data.Internal.List.Associative | {α : Type u} → {β : α → Type v} → [BEq α] → α → List ((a : α) × β a) → Option ((a : α) × β a) |
_private.Mathlib.GroupTheory.Submonoid.Inverses.0.Submonoid.leftInvEquiv._simp_2 | Mathlib.GroupTheory.Submonoid.Inverses | ∀ {α : Type u} [inst : Monoid α] {u : αˣ} {a : α}, (↑u⁻¹ = a) = (↑u * a = 1) |
Lean.Parser.Term.leading_parser._regBuiltin.Lean.Parser.Term.withAnonymousAntiquot.parenthesizer_19 | Lean.Parser.Term | IO Unit |
Int.fib_neg_one | Mathlib.Data.Int.Fib.Basic | Int.fib (-1) = 1 |
MonadReader.casesOn | Init.Prelude | {ρ : Type u} →
{m : Type u → Type v} →
{motive : MonadReader ρ m → Sort u_1} → (t : MonadReader ρ m) → ((read : m ρ) → motive { read := read }) → motive t |
NumberField.instIsAlgebraicSubtypeMemSubfield | Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | ∀ {K : Type u_2} [inst : Field K] [inst_1 : CharZero K] [Algebra.IsAlgebraic ℚ K] (k : Subfield K),
Algebra.IsAlgebraic (↥k) K |
Set.graphOn_singleton | Mathlib.Data.Set.Prod | ∀ {α : Type u_1} {β : Type u_2} (f : α → β) (x : α), Set.graphOn f {x} = {(x, f x)} |
_private.Mathlib.NumberTheory.Padics.PadicNumbers.0.Padic.norm_intCast_eq_one_iff._simp_1_3 | Mathlib.NumberTheory.Padics.PadicNumbers | ∀ {m n : ℤ}, IsCoprime m n = (m.gcd n = 1) |
mul_le_mul_left_of_neg._simp_1 | Mathlib.Algebra.Order.Ring.Unbundled.Basic | ∀ {R : Type u} [inst : Semiring R] [inst_1 : LinearOrder R] [ExistsAddOfLE R] [PosMulStrictMono R] [AddRightMono R]
[AddRightReflectLE R] {a b c : R}, c < 0 → (c * a ≤ c * b) = (b ≤ a) |
linearIndependent_fin_succ | Mathlib.LinearAlgebra.LinearIndependent.Lemmas | ∀ {K : Type u_3} {V : Type u} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {n : ℕ}
{v : Fin (n + 1) → V},
LinearIndependent K v ↔ LinearIndependent K (Fin.tail v) ∧ v 0 ∉ Submodule.span K (Set.range (Fin.tail v)) |
subset_affineSpan | Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | ∀ (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 : Set P), s ⊆ ↑(affineSpan k s) |
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