name stringlengths 2 347 | module stringlengths 6 90 | type stringlengths 1 5.42M |
|---|---|---|
Lean.Elab.Tactic.BVDecide.Frontend.SolverMode._sizeOf_1 | Std.Tactic.BVDecide.Syntax | Lean.Elab.Tactic.BVDecide.Frontend.SolverMode → ℕ |
Equiv.addEquiv._proof_1 | Mathlib.Algebra.Group.TransferInstance | ∀ {α : Type u_2} {β : Type u_1} (e : α ≃ β) [inst : Add β] (x y : α), e.toFun (x + y) = e.toFun x + e.toFun y |
SupIrred.ne_bot | Mathlib.Order.Irreducible | ∀ {α : Type u_2} [inst : SemilatticeSup α] {a : α} [inst_1 : OrderBot α], SupIrred a → a ≠ ⊥ |
HomologicalComplex.mapBifunctor₂₃.d₃_eq | Mathlib.Algebra.Homology.BifunctorAssociator | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₂₃ : Type u_4} {C₃ : Type u_5} {C₄ : Type u_6}
[inst : CategoryTheory.Category.{v_1, u_1} C₁] [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂]
[inst_2 : CategoryTheory.Category.{v_3, u_5} C₃] [inst_3 : CategoryTheory.Category.{v_4, u_6} C₄]
[inst_4 : CategoryTheory.Category.{v_6, u_4} C₂₃] [inst_5 : CategoryTheory.Limits.HasZeroMorphisms C₁]
[inst_6 : CategoryTheory.Limits.HasZeroMorphisms C₂] [inst_7 : CategoryTheory.Limits.HasZeroMorphisms C₃]
[inst_8 : CategoryTheory.Preadditive C₂₃] [inst_9 : CategoryTheory.Preadditive C₄]
(F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂₃ C₄))
(G₂₃ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₂₃)) [inst_10 : G₂₃.PreservesZeroMorphisms]
[inst_11 : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms] [inst_12 : F.PreservesZeroMorphisms]
[inst_13 : ∀ (X₁ : C₁), (F.obj X₁).Additive] {ι₁ : Type u_7} {ι₂ : Type u_8} {ι₃ : Type u_9} {ι₁₂ : Type u_10}
{ι₂₃ : Type u_11} {ι₄ : Type u_12} [inst_14 : DecidableEq ι₄] {c₁ : ComplexShape ι₁} {c₂ : ComplexShape ι₂}
{c₃ : ComplexShape ι₃} (K₁ : HomologicalComplex C₁ c₁) (K₂ : HomologicalComplex C₂ c₂) (K₃ : HomologicalComplex C₃ c₃)
(c₁₂ : ComplexShape ι₁₂) (c₂₃ : ComplexShape ι₂₃) (c₄ : ComplexShape ι₄) [inst_15 : TotalComplexShape c₁ c₂ c₁₂]
[inst_16 : TotalComplexShape c₁₂ c₃ c₄] [inst_17 : TotalComplexShape c₂ c₃ c₂₃]
[inst_18 : TotalComplexShape c₁ c₂₃ c₄] [inst_19 : K₂.HasMapBifunctor K₃ G₂₃ c₂₃]
[inst_20 : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄] [inst_21 : DecidableEq ι₂₃]
[inst_22 : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄] (i₁ : ι₁) (i₂ : ι₂) {i₃ i₃' : ι₃},
c₃.Rel i₃ i₃' → ∀ (j : ι₄), ⋯ = ⋯ |
Set.nonempty_sInter._simp_1 | Mathlib.Data.Set.Lattice | ∀ {α : Type u_1} {c : Set (Set α)}, (⋂₀ c).Nonempty = ∃ a, ∀ b ∈ c, a ∈ b |
_private.Mathlib.RingTheory.Valuation.ValuationSubring.0.ValuationSubring.ofPrime_idealOfLE._simp_1_2 | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {M : Type u_1} [inst : MulOneClass M] {s : Subsemigroup M} {x : M} (h_one : 1 ∈ s.carrier),
(x ∈ { toSubsemigroup := s, one_mem' := h_one }) = (x ∈ s) |
DirSupInaccOn | Mathlib.Topology.Order.ScottTopology | {α : Type u_1} → [Preorder α] → Set (Set α) → Set α → Prop |
extDeriv_apply_vectorField_of_pairwise_commute | Mathlib.Analysis.Calculus.DifferentialForm.VectorField | ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : ℕ} {x : E}
{ω : E → E [⋀^Fin n]→L[𝕜] F} {V : Fin (n + 1) → E → E},
DifferentiableAt 𝕜 ω x →
(∀ (i : Fin (n + 1)), DifferentiableAt 𝕜 (V i) x) →
(Pairwise fun i j => VectorField.lieBracket 𝕜 (V i) (V j) x = 0) →
((extDeriv ω x) fun x_1 => V x_1 x) =
∑ i, (-1) ^ ↑i • (fderiv 𝕜 (fun x => (ω x) (i.removeNth fun x_1 => V x_1 x)) x) (V i x) |
BitVec.toNat_cpop_concat | Init.Data.BitVec.Lemmas | ∀ {w : ℕ} {x : BitVec w} {b : Bool}, (x.concat b).cpop.toNat = b.toNat + x.cpop.toNat |
Polynomial.recOnHorner._unary._proof_15 | Mathlib.Algebra.Polynomial.Inductions | ∀ {R : Type u_2} [inst : Semiring R] {M : Polynomial R → Sort u_1} (p : Polynomial R),
M (p.divX * Polynomial.X + Polynomial.C 0) = M (p.divX * Polynomial.X + 0) |
Aesop.instInhabitedNormalizationState.default | Aesop.Tree.Data | Aesop.NormalizationState |
_private.Std.Data.DHashMap.Lemmas.0.Std.DHashMap.mem_alter._simp_1_1 | Std.Data.DHashMap.Lemmas | ∀ {α : Type u} {β : α → Type v} {x : BEq α} {x_1 : Hashable α} {m : Std.DHashMap α β} {a : α},
(a ∈ m) = (m.contains a = true) |
DifferentiableOn.mul_const | Mathlib.Analysis.Calculus.FDeriv.Mul | ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] {s : Set E} {𝔸 : Type u_5} [inst_3 : NormedRing 𝔸] [inst_4 : NormedAlgebra 𝕜 𝔸]
{a : E → 𝔸}, DifferentiableOn 𝕜 a s → ∀ (b : 𝔸), DifferentiableOn 𝕜 (fun y => a y * b) s |
CategoryTheory.Limits.KernelFork.IsLimit.ofιUnop._proof_4 | Mathlib.CategoryTheory.Limits.Shapes.Opposites.Kernels | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{K X Y : Cᵒᵖ} (i : K ⟶ X) {f : X ⟶ Y} (w : CategoryTheory.CategoryStruct.comp i f = 0)
(h : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι i w)) {Z' : C} (x : Opposite.unop X ⟶ Z')
(x_1 : CategoryTheory.CategoryStruct.comp f.unop x = 0),
CategoryTheory.CategoryStruct.comp i.unop (h.lift (CategoryTheory.Limits.KernelFork.ofι x.op ⋯)).unop = x |
FiniteField.frobeniusAlgEquiv._proof_1 | Mathlib.FieldTheory.Finite.Basic | ∀ (K : Type u_2) (R : Type u_1) [inst : Field K] [inst_1 : Fintype K] [inst_2 : CommRing R] [inst_3 : Algebra K R]
(p : ℕ) [ExpChar R p] [PerfectRing R p], Function.Bijective ⇑(FiniteField.frobeniusAlgHom K R) |
UniformSpace.Completion.extensionHom._proof_2 | Mathlib.Topology.Algebra.UniformRing | ∀ {α : Type u_2} [inst : Ring α] {β : Type u_1} [inst_1 : Ring β], AddMonoidHomClass (α →+* β) α β |
_private.Batteries.Data.List.Lemmas.0.List.pos_findIdxNth_getElem._proof_1_12 | Batteries.Data.List.Lemmas | ∀ {α : Type u_1} {p : α → Bool} (tail : List α) {n : ℕ},
List.findIdxNth p tail (n - 1) + 1 ≤ tail.length → List.findIdxNth p tail (n - 1) < tail.length |
_private.Lean.Data.FuzzyMatching.0.Lean.FuzzyMatching.Score | Lean.Data.FuzzyMatching | Type |
_private.Qq.Macro.0.Qq.Impl.quoteExpr.match_1 | Qq.Macro | (motive : Qq.Impl.ExprBackSubstResult → Sort u_1) →
(r : Qq.Impl.ExprBackSubstResult) →
((r : Lean.Expr) → motive (Qq.Impl.ExprBackSubstResult.quoted r)) →
((r : Lean.Expr) → motive (Qq.Impl.ExprBackSubstResult.unquoted r)) → motive r |
FreeLieAlgebra.lift_of_apply | Mathlib.Algebra.Lie.Free | ∀ {R : Type u} {X : Type v} [inst : CommRing R] {L : Type w} [inst_1 : LieRing L] [inst_2 : LieAlgebra R L] (f : X → L)
(x : X), ((FreeLieAlgebra.lift R) f) (FreeLieAlgebra.of R x) = f x |
CategoryTheory.SplitMono | Mathlib.CategoryTheory.EpiMono | {C : Type u₁} → [inst : CategoryTheory.Category.{v₁, u₁} C] → {X Y : C} → (X ⟶ Y) → Type v₁ |
MDifferentiableWithinAt.prodMap' | Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | ∀ {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] [inst_5 : ChartedSpace H M] {E' : Type u_5} [inst_6 : NormedAddCommGroup E']
[inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
{M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {F : Type u_11}
[inst_11 : NormedAddCommGroup F] [inst_12 : NormedSpace 𝕜 F] {G : Type u_12} [inst_13 : TopologicalSpace G]
{J : ModelWithCorners 𝕜 F G} {N : Type u_13} [inst_14 : TopologicalSpace N] [inst_15 : ChartedSpace G N]
{F' : Type u_14} [inst_16 : NormedAddCommGroup F'] [inst_17 : NormedSpace 𝕜 F'] {G' : Type u_15}
[inst_18 : TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [inst_19 : TopologicalSpace N']
[inst_20 : ChartedSpace G' N'] {s : Set M} {f : M → M'} {g : N → N'} {r : Set N} {p : M × N},
MDiffAt[s] f p.1 → MDiffAt[r] g p.2 → MDiffAt[s ×ˢ r] (Prod.map f g) p |
CategoryTheory.Localization.Construction.morphismProperty_eq_top' | Mathlib.CategoryTheory.Localization.Construction | ∀ {C : Type uC} [inst : CategoryTheory.Category.{uC', uC} C] {W : CategoryTheory.MorphismProperty C}
(P : CategoryTheory.MorphismProperty W.Localization) [P.IsStableUnderComposition],
(∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)) → (∀ ⦃X Y : W.Localization⦄ (e : X ≅ Y), P e.hom → P e.inv) → P = ⊤ |
FreeGroup.of_ne_one._simp_2 | Mathlib.GroupTheory.FreeGroup.Reduce | ∀ {α : Type u_1} (a : α), (FreeGroup.of a = 1) = False |
Lean.TSyntax.ctorIdx | Init.Prelude | {ks : Lean.SyntaxNodeKinds} → Lean.TSyntax ks → ℕ |
AddEquiv.toMultiplicativeLeft._proof_7 | Mathlib.Algebra.Group.Equiv.TypeTags | ∀ {G : Type u_1} {H : Type u_2} [inst : AddZeroClass G] [inst_1 : MulOneClass H] (f : Multiplicative G ≃* H),
Function.RightInverse f.invFun f.toFun |
String.Pos.Raw.instLTCiOfNatInt | Init.Data.String.OrderInstances | Lean.Grind.ToInt.LT String.Pos.Raw (Lean.Grind.IntInterval.ci 0) |
Std.DTreeMap.Internal.Impl.Const.get?_congr | Std.Data.DTreeMap.Internal.Lemmas | ∀ {α : Type u} {instOrd : Ord α} {β : Type v} {t : Std.DTreeMap.Internal.Impl α fun x => β} [Std.TransOrd α],
t.WF →
∀ {a b : α},
compare a b = Ordering.eq → Std.DTreeMap.Internal.Impl.Const.get? t a = Std.DTreeMap.Internal.Impl.Const.get? t b |
_private.Mathlib.Topology.Baire.LocallyCompactRegular.0.BaireSpace.of_t2Space_locallyCompactSpace._simp_2 | Mathlib.Topology.Baire.LocallyCompactRegular | ∀ {α : Type u} (x : α) (a b : Set α), (x ∈ a ∩ b) = (x ∈ a ∧ x ∈ b) |
Matrix.detp_smul_adjp | Mathlib.LinearAlgebra.Matrix.SemiringInverse | ∀ {n : Type u_1} {R : Type u_3} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommSemiring R]
{A B : Matrix n n R},
A * B = 1 →
A + (Matrix.detp 1 A • Matrix.adjp (-1) B + Matrix.detp (-1) A • Matrix.adjp 1 B) =
Matrix.detp 1 A • Matrix.adjp 1 B + Matrix.detp (-1) A • Matrix.adjp (-1) B |
Std.DHashMap.Internal.AssocList.foldrM | Std.Data.DHashMap.Internal.AssocList.Basic | {α : Type u} →
{β : α → Type v} →
{δ : Type w} →
{m : Type w → Type w'} → [Monad m] → ((a : α) → β a → δ → m δ) → δ → Std.DHashMap.Internal.AssocList α β → m δ |
CategoryTheory.Limits.isCokernelEpiComp._proof_1 | Mathlib.CategoryTheory.Limits.Shapes.Kernels | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{X Y : C} {f : X ⟶ Y} {c : CategoryTheory.Limits.CokernelCofork f} {W : C} (g : W ⟶ X) {h : W ⟶ Y},
h = CategoryTheory.CategoryStruct.comp g f →
CategoryTheory.CategoryStruct.comp h (CategoryTheory.Limits.Cofork.π c) = 0 |
_private.Mathlib.Analysis.CStarAlgebra.Multiplier.0.DoubleCentralizer.instCStarRing._simp_2 | Mathlib.Analysis.CStarAlgebra.Multiplier | ∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : DenselyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂]
(f : E →SL[σ₁₂] F), ‖f‖₊ = sSup ((fun x => ‖f x‖₊) '' Metric.closedBall 0 1) |
CategoryTheory.Limits.isIsoZeroEquiv._proof_3 | Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(X Y : C),
CategoryTheory.CategoryStruct.id X = 0 ∧ CategoryTheory.CategoryStruct.id Y = 0 →
CategoryTheory.CategoryStruct.comp 0 0 = CategoryTheory.CategoryStruct.id X ∧
CategoryTheory.CategoryStruct.comp 0 0 = CategoryTheory.CategoryStruct.id Y |
_private.Std.Time.Format.Basic.0.Std.Time.leftPad | Std.Time.Format.Basic | ℕ → Char → String → String |
_private.Mathlib.LinearAlgebra.Matrix.FixedDetMatrices.0.FixedDetMatrices.reduce_mem_reps._simp_1_6 | Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α] [AddLeftMono α] [AddRightMono α] {a b : α}, (b ≤ -a) = (a ≤ -b) |
_private.Mathlib.Tactic.Linter.Style.0.Mathlib.Linter.Style.longLine.longLineLinter | Mathlib.Tactic.Linter.Style | Lean.Linter |
SemimoduleCat.Hom._sizeOf_1 | Mathlib.Algebra.Category.ModuleCat.Semi | {R : Type u} → {inst : Semiring R} → {M N : SemimoduleCat R} → [SizeOf R] → M.Hom N → ℕ |
UInt16.fromExpr | Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt | Lean.Expr → Lean.Meta.SimpM (Option UInt16) |
InfHom.id.eq_1 | Mathlib.Order.Hom.Lattice | ∀ (α : Type u_2) [inst : Min α], InfHom.id α = { toFun := id, map_inf' := ⋯ } |
Action.instConcreteCategoryHomSubtypeV | Mathlib.CategoryTheory.Action.Basic | (V : Type u_1) →
[inst : CategoryTheory.Category.{v_1, u_1} V] →
(G : Type u_2) →
[inst_1 : Monoid G] →
{FV : V → V → Type u_3} →
{CV : V → Type u_4} →
[inst_2 : (X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] →
[inst_3 : CategoryTheory.ConcreteCategory V FV] →
CategoryTheory.ConcreteCategory (Action V G) (Action.HomSubtype V G) |
SemidirectProduct.inr_splitting | Mathlib.GroupTheory.GroupExtension.Defs | {N : Type u_1} →
{G : Type u_3} →
[inst : Group G] → [inst_1 : Group N] → (φ : G →* MulAut N) → (SemidirectProduct.toGroupExtension φ).Splitting |
TensorAlgebra.GradedAlgebra.ι_apply._proof_1 | Mathlib.LinearAlgebra.TensorAlgebra.Grading | ∀ (R : Type u_1) (M : Type u_2) [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (m : M),
(TensorAlgebra.ι R) m ∈ (TensorAlgebra.ι R).range ^ 1 |
CartanMatrix.E₈ | Mathlib.Data.Matrix.Cartan | Matrix (Fin 8) (Fin 8) ℤ |
Mathlib.Tactic.Translate.Config.doc._default | Mathlib.Tactic.Translate.Core | Option String |
_private.Mathlib.Combinatorics.SetFamily.AhlswedeZhang.0.Finset.infs_aux | Mathlib.Combinatorics.SetFamily.AhlswedeZhang | ∀ {α : Type u_1} [inst : DistribLattice α] [inst_1 : DecidableEq α] {s t : Finset α} {a : α},
a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t |
PowerSeries.instInhabited | Mathlib.RingTheory.PowerSeries.Basic | {R : Type u_1} → [Inhabited R] → Inhabited (PowerSeries R) |
NonAssocRing.toAddCommGroupWithOne | Mathlib.Algebra.Ring.Defs | {α : Type u_1} → [self : NonAssocRing α] → AddCommGroupWithOne α |
ContDiffWithinAt.contDiffBump | Mathlib.Analysis.Calculus.BumpFunction.Basic | ∀ {E : Type u_1} {X : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup X]
[inst_3 : NormedSpace ℝ X] [inst_4 : HasContDiffBump E] {n : ℕ∞} {c g : X → E} {s : Set X}
{f : (x : X) → ContDiffBump (c x)} {x : X},
ContDiffWithinAt ℝ (↑n) c s x →
ContDiffWithinAt ℝ (↑n) (fun x => (f x).rIn) s x →
ContDiffWithinAt ℝ (↑n) (fun x => (f x).rOut) s x →
ContDiffWithinAt ℝ (↑n) g s x → ContDiffWithinAt ℝ (↑n) (fun x => ↑(f x) (g x)) s x |
WithCStarModule.norm_apply_le_norm | 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)] [StarOrderedRing A]
(x : WithCStarModule A ((i : ι) → E i)) (i : ι), ‖x i‖ ≤ ‖x‖ |
Nat.xor_right_injective | Batteries.Data.Nat.Bitwise | ∀ {x : ℕ}, Function.Injective fun x_1 => x ^^^ x_1 |
TopologicalSpace.le_def | Mathlib.Topology.Order | ∀ {α : Type u_1} {t s : TopologicalSpace α}, t ≤ s ↔ IsOpen ≤ IsOpen |
String.Slice.Pattern.Model.SlicesFrom.extend | Init.Data.String.Lemmas.Pattern.Split | {s : String.Slice} →
(p₁ : s.Pos) →
{p₂ : s.Pos} → p₁ ≤ p₂ → String.Slice.Pattern.Model.SlicesFrom p₂ → String.Slice.Pattern.Model.SlicesFrom p₁ |
List.destutter'_of_chain | Mathlib.Data.List.Destutter | ∀ {α : Type u_1} (l : List α) (R : α → α → Prop) [inst : DecidableRel R] {a : α},
List.IsChain R (a :: l) → List.destutter' R a l = a :: l |
ValuationSubring.one_mem | Mathlib.RingTheory.Valuation.ValuationSubring | ∀ {K : Type u} [inst : Field K] (A : ValuationSubring K), 1 ∈ A |
TrivSqZeroExt.instAlgebra._proof_2 | Mathlib.Algebra.TrivSqZeroExt.Basic | ∀ (R' : Type u_1) (M : Type u_2) [inst : CommSemiring R'] [inst_1 : AddCommMonoid M] [inst_2 : Module R' M],
IsScalarTower R' R' M |
Lean.Elab.Command.InductiveElabStep3.finalize | Lean.Elab.MutualInductive | Lean.Elab.Command.InductiveElabStep3 → Lean.Elab.TermElabM Unit |
CategoryTheory.PullbackShift.adjunction | Mathlib.CategoryTheory.Shift.Pullback | {C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
{A : Type u_2} →
{B : Type u_3} →
[inst_1 : AddMonoid A] →
[inst_2 : AddMonoid B] →
(φ : A →+ B) →
[inst_3 : CategoryTheory.HasShift C B] →
{D : Type u_4} →
[inst_4 : CategoryTheory.Category.{v_2, u_4} D] →
[inst_5 : CategoryTheory.HasShift D B] →
{F : CategoryTheory.Functor C D} →
{G : CategoryTheory.Functor D C} →
(F ⊣ G) →
(CategoryTheory.PullbackShift.functor φ F ⊣ CategoryTheory.PullbackShift.functor φ G) |
MeasureTheory.SimpleFunc.ofIsEmpty._proof_1 | Mathlib.MeasureTheory.Function.SimpleFunc | ∀ {α : Type u_1} [IsEmpty α], Finite α |
Turing.TM0.Machine.map_step | Mathlib.Computability.TuringMachine.PostTuringMachine | ∀ {Γ : Type u_1} [inst : Inhabited Γ] {Γ' : Type u_2} [inst_1 : Inhabited Γ'] {Λ : Type u_3} [inst_2 : Inhabited Λ]
{Λ' : Type u_4} [inst_3 : Inhabited Λ'] (M : Turing.TM0.Machine Γ Λ) (f₁ : Turing.PointedMap Γ Γ')
(f₂ : Turing.PointedMap Γ' Γ) (g₁ : Λ → Λ') (g₂ : Λ' → Λ) {S : Set Λ},
Function.RightInverse f₁.f f₂.f →
(∀ q ∈ S, g₂ (g₁ q) = q) →
∀ (c : Turing.TM0.Cfg Γ Λ),
c.q ∈ S →
Option.map (Turing.TM0.Cfg.map f₁ g₁) (Turing.TM0.step M c) =
Turing.TM0.step (M.map f₁ f₂ g₁ g₂) (Turing.TM0.Cfg.map f₁ g₁ c) |
CategoryTheory.NatTrans.CommShift.verticalComposition | Mathlib.CategoryTheory.Shift.CommShift | ∀ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {D₁ : Type u_4} {D₂ : Type u_5} {D₃ : Type u_6}
[inst : CategoryTheory.Category.{v_1, u_1} C₁] [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂]
[inst_2 : CategoryTheory.Category.{v_3, u_3} C₃] [inst_3 : CategoryTheory.Category.{v_4, u_4} D₁]
[inst_4 : CategoryTheory.Category.{v_5, u_5} D₂] [inst_5 : CategoryTheory.Category.{v_6, u_6} D₃]
{F₁₂ : CategoryTheory.Functor C₁ C₂} {F₂₃ : CategoryTheory.Functor C₂ C₃} {F₁₃ : CategoryTheory.Functor C₁ C₃}
(α : F₁₃ ⟶ F₁₂.comp F₂₃) {G₁₂ : CategoryTheory.Functor D₁ D₂} {G₂₃ : CategoryTheory.Functor D₂ D₃}
{G₁₃ : CategoryTheory.Functor D₁ D₃} (β : G₁₂.comp G₂₃ ⟶ G₁₃) {L₁ : CategoryTheory.Functor C₁ D₁}
{L₂ : CategoryTheory.Functor C₂ D₂} {L₃ : CategoryTheory.Functor C₃ D₃} (e₁₂ : F₁₂.comp L₂ ⟶ L₁.comp G₁₂)
(e₂₃ : F₂₃.comp L₃ ⟶ L₂.comp G₂₃) (e₁₃ : F₁₃.comp L₃ ⟶ L₁.comp G₁₃) (A : Type u_7) [inst_6 : AddMonoid A]
[inst_7 : CategoryTheory.HasShift C₁ A] [inst_8 : CategoryTheory.HasShift C₂ A]
[inst_9 : CategoryTheory.HasShift C₃ A] [inst_10 : CategoryTheory.HasShift D₁ A]
[inst_11 : CategoryTheory.HasShift D₂ A] [inst_12 : CategoryTheory.HasShift D₃ A] [inst_13 : F₁₂.CommShift A]
[inst_14 : F₂₃.CommShift A] [inst_15 : F₁₃.CommShift A] [CategoryTheory.NatTrans.CommShift α A]
[inst_17 : G₁₂.CommShift A] [inst_18 : G₂₃.CommShift A] [inst_19 : G₁₃.CommShift A]
[CategoryTheory.NatTrans.CommShift β A] [inst_21 : L₁.CommShift A] [inst_22 : L₂.CommShift A]
[inst_23 : L₃.CommShift A] [CategoryTheory.NatTrans.CommShift e₁₂ A] [CategoryTheory.NatTrans.CommShift e₂₃ A],
e₁₃ =
CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight α L₃)
(CategoryTheory.CategoryStruct.comp ⋯ ⋯) →
CategoryTheory.NatTrans.CommShift e₁₃ A |
CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.casesOn | Mathlib.CategoryTheory.Monoidal.DayConvolution | {C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{V : Type u₂} →
[inst_1 : CategoryTheory.Category.{v₂, u₂} V] →
[inst_2 : CategoryTheory.MonoidalCategory C] →
[inst_3 : CategoryTheory.MonoidalCategory V] →
{D : Type u₃} →
[inst_4 : CategoryTheory.Category.{v₃, u₃} D] →
[inst_5 : CategoryTheory.MonoidalCategoryStruct D] →
{motive : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D → Sort u} →
(t : CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct C V D) →
((ι : CategoryTheory.Functor D (CategoryTheory.Functor C V)) →
(convolutionExtensionUnit :
(d d' : D) →
CategoryTheory.MonoidalCategory.externalProduct (ι.obj d) (ι.obj d') ⟶
(CategoryTheory.MonoidalCategory.tensor C).comp
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))) →
(isPointwiseLeftKanExtensionConvolutionExtensionUnit :
(d d' : D) →
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj d d'))
(convolutionExtensionUnit d d')).IsPointwiseLeftKanExtension) →
(unitUnit :
CategoryTheory.MonoidalCategoryStruct.tensorUnit V ⟶
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D)).obj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) →
(isPointwiseLeftKanExtensionUnitUnit :
(CategoryTheory.Functor.LeftExtension.mk
(ι.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit D))
{ app := fun x => unitUnit, naturality := ⋯ }).IsPointwiseLeftKanExtension) →
(faithful_ι : ι.Faithful) →
(convolutionExtensionUnit_comp_ι_map_tensorHom_app :
∀ {d₁ d₂ d₁' d₂' : D} (f₁ : d₁ ⟶ d₁') (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.tensorHom f₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.tensorHom ((ι.map f₁).app x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁' d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :
∀ (d₁ : D) {d₂ d₂' : D} (f₂ : d₂ ⟶ d₂') (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map (CategoryTheory.MonoidalCategoryStruct.whiskerLeft d₁ f₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft ((ι.obj d₁).obj x)
((ι.map f₂).app y))
((convolutionExtensionUnit d₁ d₂').app (x, y))) →
(convolutionExtensionUnit_comp_ι_map_whiskerRight_app :
∀ {d₁ d₁' : D} (f₁ : d₁ ⟶ d₁') (d₂ : D) (x y : C),
CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d₁ d₂).app (x, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f₁ d₂)).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y)) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight ((ι.map f₁).app x)
((ι.obj d₂).obj y))
((convolutionExtensionUnit d₁' d₂).app (x, y))) →
(associator_hom_unit_unit :
∀ (d d' d'' : D) (x y z : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight
((convolutionExtensionUnit d d').app (x, y)) ((ι.obj d'').obj z))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorObj d d')
d'').app
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y, z))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.associator d d'
d'').hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorObj x y) z))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.associator ((ι.obj d).obj x)
((ι.obj d', ι.obj d'').1.obj (y, z).1)
((ι.obj d', ι.obj d'').2.obj (y, z).2)).hom
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj x)
((convolutionExtensionUnit d' d'').app (y, z)))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d'')).app
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z))
((ι.obj
(CategoryTheory.MonoidalCategoryStruct.tensorObj d
(CategoryTheory.MonoidalCategoryStruct.tensorObj d'
d''))).map
(CategoryTheory.MonoidalCategoryStruct.associator
(x, CategoryTheory.MonoidalCategoryStruct.tensorObj y z).1
y z).inv)))) →
(leftUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerRight unitUnit
((ι.obj d).obj y))
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit
(CategoryTheory.MonoidalCategoryStruct.tensorUnit D)
d).app
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C, y))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C) y))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.leftUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.leftUnitor y).inv)) →
(rightUnitor_hom_unit_app :
∀ (d : D) (y : C),
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft
((ι.obj d).obj y) unitUnit)
(CategoryTheory.CategoryStruct.comp
((convolutionExtensionUnit d
(CategoryTheory.MonoidalCategoryStruct.tensorUnit
D)).app
(y, CategoryTheory.MonoidalCategoryStruct.tensorUnit C))
((ι.map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
d).hom).app
(CategoryTheory.MonoidalCategoryStruct.tensorObj y
(CategoryTheory.MonoidalCategoryStruct.tensorUnit C)))) =
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategoryStruct.rightUnitor
((ι.obj d).obj y)).hom
((ι.obj d).map
(CategoryTheory.MonoidalCategoryStruct.rightUnitor y).inv)) →
motive
{ ι := ι, convolutionExtensionUnit := convolutionExtensionUnit,
isPointwiseLeftKanExtensionConvolutionExtensionUnit :=
isPointwiseLeftKanExtensionConvolutionExtensionUnit,
unitUnit := unitUnit,
isPointwiseLeftKanExtensionUnitUnit :=
isPointwiseLeftKanExtensionUnitUnit,
faithful_ι := faithful_ι,
convolutionExtensionUnit_comp_ι_map_tensorHom_app :=
convolutionExtensionUnit_comp_ι_map_tensorHom_app,
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app :=
convolutionExtensionUnit_comp_ι_map_whiskerLeft_app,
convolutionExtensionUnit_comp_ι_map_whiskerRight_app :=
convolutionExtensionUnit_comp_ι_map_whiskerRight_app,
associator_hom_unit_unit := associator_hom_unit_unit,
leftUnitor_hom_unit_app := leftUnitor_hom_unit_app,
rightUnitor_hom_unit_app := rightUnitor_hom_unit_app }) →
motive t |
Lean.Json.instCoeArrayStructured | Lean.Data.Json.Basic | Coe (Array Lean.Json) Lean.Json.Structured |
groupCohomology.map_one_fst_of_isCocycle₂ | Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | ∀ {G : Type u_1} {A : Type u_2} [inst : Monoid G] [inst_1 : AddCommGroup A] [inst_2 : MulAction G A] {f : G × G → A},
groupCohomology.IsCocycle₂ f → ∀ (g : G), f (1, g) = f (1, 1) |
_private.Mathlib.MeasureTheory.VectorMeasure.AddContent.0.MeasureTheory.VectorMeasure.exists_extension_of_isSetRing_of_le_measure_of_dense._simp_1_6 | Mathlib.MeasureTheory.VectorMeasure.AddContent | ∀ {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α},
MeasurableSet s₁ → MeasurableSet s₂ → MeasurableSet (s₁ ∪ s₂) = True |
Ordinal.iterate_veblen_lt_gamma_zero | Mathlib.SetTheory.Ordinal.Veblen | ∀ (n : ℕ), (fun a => Ordinal.veblen a 0)^[n] 0 < Ordinal.gamma 0 |
GaloisCoinsertion.isAtom_of_image | Mathlib.Order.Atoms | ∀ {α : Type u_2} {β : Type u_3} [inst : PartialOrder α] [inst_1 : PartialOrder β] [inst_2 : OrderBot α]
[inst_3 : OrderBot β] {l : α → β} {u : β → α} (gi : GaloisCoinsertion l u) {a : α}, IsAtom (l a) → IsAtom a |
Mathlib.Tactic.Widget.StringDiagram.Kind.monoidal.sizeOf_spec | Mathlib.Tactic.Widget.StringDiagram | sizeOf Mathlib.Tactic.Widget.StringDiagram.Kind.monoidal = 1 |
_private.Lean.Meta.Tactic.Contradiction.0.Lean.Meta.isGenDiseq | Lean.Meta.Tactic.Contradiction | Lean.Expr → Bool |
DifferentiableOn.sinh | Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : E → ℝ} {s : Set E},
DifferentiableOn ℝ f s → DifferentiableOn ℝ (fun x => Real.sinh (f x)) s |
Orientation.inner_smul_rotation_pi_div_two_smul_right | Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | ∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : Fact (Module.finrank ℝ V = 2)]
(o : Orientation ℝ V (Fin 2)) (x : V) (r₁ r₂ : ℝ), inner ℝ (r₂ • x) (r₁ • (o.rotation ↑(Real.pi / 2)) x) = 0 |
TopCat.Sheaf.interUnionPullbackCone._proof_3 | Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | ∀ {X : TopCat} (U V : TopologicalSpace.Opens ↑X), U ⊓ V ≤ V |
Commute.zpow_right | Mathlib.Algebra.Group.Commute.Basic | ∀ {G : Type u_1} [inst : Group G] {a b : G}, Commute a b → ∀ (m : ℤ), Commute a (b ^ m) |
Filter.IsCobounded.mk | Mathlib.Order.Filter.IsBounded | ∀ {α : Type u_1} {r : α → α → Prop} {f : Filter α} [IsTrans α r] (a : α),
(∀ s ∈ f, ∃ x ∈ s, r a x) → Filter.IsCobounded r f |
SSet.stdSimplex.spineId | Mathlib.AlgebraicTopology.SimplicialSet.Path | (n : ℕ) → (SSet.stdSimplex.obj (SimplexCategory.mk n)).Path n |
instSemilatticeSupENNReal | Mathlib.Data.ENNReal.Basic | SemilatticeSup ENNReal |
Polynomial.Nontrivial.of_polynomial_ne | Mathlib.Algebra.Polynomial.Basic | ∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R}, p ≠ q → Nontrivial R |
Subfield.instIsScalarTowerSubtypeMem | Mathlib.Algebra.Field.Subfield.Basic | ∀ {K : Type u} [inst : DivisionRing K] {X : Type u_1} {Y : Type u_2} [inst_1 : SMul X Y] [inst_2 : SMul K X]
[inst_3 : SMul K Y] [IsScalarTower K X Y] (F : Subfield K), IsScalarTower (↥F) X Y |
AlgebraicGeometry.Scheme.Hom.mem_smoothLocus | Mathlib.AlgebraicGeometry.Morphisms.Smooth | ∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [inst : AlgebraicGeometry.LocallyOfFinitePresentation f] {x : ↥X},
x ∈ AlgebraicGeometry.Scheme.Hom.smoothLocus f ↔
(CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x)).FormallySmooth |
Complex.arg_exp_mul_I | Mathlib.Analysis.SpecialFunctions.Complex.Arg | ∀ (θ : ℝ), (Complex.exp (↑θ * Complex.I)).arg = toIocMod Real.two_pi_pos (-Real.pi) θ |
ContinuousMultilinearMap.smulRight | Mathlib.Topology.Algebra.Module.Multilinear.Basic | {R : Type u} →
{ι : Type v} →
{M₁ : ι → Type w₁} →
{M₂ : Type w₂} →
[inst : CommSemiring R] →
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] →
[inst_2 : AddCommMonoid M₂] →
[inst_3 : (i : ι) → Module R (M₁ i)] →
[inst_4 : Module R M₂] →
[inst_5 : TopologicalSpace R] →
[inst_6 : (i : ι) → TopologicalSpace (M₁ i)] →
[inst_7 : TopologicalSpace M₂] →
[ContinuousSMul R M₂] → ContinuousMultilinearMap R M₁ R → M₂ → ContinuousMultilinearMap R M₁ M₂ |
AdjoinRoot.liftHom_mk | Mathlib.RingTheory.AdjoinRoot | ∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] (f : Polynomial R) [inst_1 : CommRing S] {a : S}
[inst_2 : Algebra R S] (hfx : (Polynomial.aeval a) f = 0) {g : Polynomial R},
(AdjoinRoot.liftAlgHom f (Algebra.ofId R S) a hfx) ((AdjoinRoot.mk f) g) = (Polynomial.aeval a) g |
_private.Mathlib.Data.EReal.Operations.0.EReal.add_ne_top_iff_ne_top₂._simp_1_2 | Mathlib.Data.EReal.Operations | ∀ (x : ℝ), (↑x = ⊤) = False |
List.prod_mul_prod_eq_prod_zipWith_mul_prod_drop | Mathlib.Algebra.BigOperators.Group.List.Basic | ∀ {M : Type u_4} [inst : CommMonoid M] (l l' : List M),
l.prod * l'.prod =
(List.zipWith (fun x1 x2 => x1 * x2) l l').prod * (List.drop l'.length l).prod * (List.drop l.length l').prod |
Lean.ScopedEnvExtension.State.rec | Lean.ScopedEnvExtension | {σ : Type} →
{motive : Lean.ScopedEnvExtension.State σ → Sort u} →
((state : σ) →
(activeScopes : Lean.NameSet) →
(delimitsLocal : Bool) →
motive { state := state, activeScopes := activeScopes, delimitsLocal := delimitsLocal }) →
(t : Lean.ScopedEnvExtension.State σ) → motive t |
Std.LawfulOrderMin.mk | Init.Data.Order.Classes | ∀ {α : Type u} [inst : Min α] [inst_1 : LE α] [toMinEqOr : Std.MinEqOr α] [toLawfulOrderInf : Std.LawfulOrderInf α],
Std.LawfulOrderMin α |
Algebra.tensorH1CotangentOfIsLocalization._proof_2 | Mathlib.RingTheory.Etale.Kaehler | ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],
MonoidHomClass ((Algebra.Generators.self R S).toExtension.Ring →+* S) (Algebra.Generators.self R S).toExtension.Ring S |
Int.le_floor_add | Mathlib.Algebra.Order.Floor.Ring | ∀ {R : Type u_2} [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : FloorRing R] [IsStrictOrderedRing R] (a b : R),
⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋ |
Std.Internal.List.containsKey_maxKey? | 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 →
∀ {km : α}, Std.Internal.List.maxKey? l = some km → Std.Internal.List.containsKey km l = true |
Lean.Language.SnapshotBundle.mk | Lean.Language.Basic | {α : Type} →
Option (Lean.Language.SyntaxGuarded (Lean.Language.SnapshotTask α)) → IO.Promise α → Lean.Language.SnapshotBundle α |
Std.IterM.TerminationMeasures.Productive.mk.injEq | Init.Data.Iterators.Basic | ∀ {α : Type w} {m : Type w → Type w'} {β : Type w} [inst : Std.Iterator α m β] (it it_1 : Std.IterM m β),
({ it := it } = { it := it_1 }) = (it = it_1) |
CategoryTheory.PreOneHypercover.cylinderX._proof_1 | Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | ∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] {S : C} {E : CategoryTheory.PreOneHypercover S}
{F : CategoryTheory.PreOneHypercover S} (f g : E.Hom F) {i : E.I₀},
CategoryTheory.CategoryStruct.comp (f.h₀ i) (F.f (f.s₀ i)) =
CategoryTheory.CategoryStruct.comp (g.h₀ i) (F.f (g.s₀ i)) |
ContinuousMultilinearMap.compContinuousLinearMap._proof_1 | Mathlib.Topology.Algebra.Module.Multilinear.Basic | ∀ {R : Type u_5} {ι : Type u_1} {M₁ : ι → Type u_2} {M₁' : ι → Type u_4} {M₄ : Type u_3} [inst : Semiring R]
[inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : (i : ι) → AddCommMonoid (M₁' i)] [inst_3 : AddCommMonoid M₄]
[inst_4 : (i : ι) → Module R (M₁ i)] [inst_5 : (i : ι) → Module R (M₁' i)] [inst_6 : Module R M₄]
[inst_7 : (i : ι) → TopologicalSpace (M₁ i)] [inst_8 : (i : ι) → TopologicalSpace (M₁' i)]
[inst_9 : TopologicalSpace M₄] (g : ContinuousMultilinearMap R M₁' M₄) (f : (i : ι) → M₁ i →L[R] M₁' i),
Continuous (g.toFun ∘ fun x i => ↑(f i) (x i)) |
CategoryTheory.Bicategory.prod._proof_22 | Mathlib.CategoryTheory.Bicategory.Product | ∀ (B : Type u_1) [inst : CategoryTheory.Bicategory B] (C : Type u_2) [inst_1 : CategoryTheory.Bicategory C]
{a b c : B × C} (f : a ⟶ b) (g : b ⟶ c),
CategoryTheory.CategoryStruct.comp
((CategoryTheory.Bicategory.associator f.1 (CategoryTheory.CategoryStruct.id b).1 g.1).prod
(CategoryTheory.Bicategory.associator f.2 (CategoryTheory.CategoryStruct.id b).2 g.2)).hom
(CategoryTheory.Prod.mkHom
(CategoryTheory.Bicategory.whiskerLeft f.1
((CategoryTheory.Bicategory.leftUnitor g.1).prod (CategoryTheory.Bicategory.leftUnitor g.2)).hom.1)
(CategoryTheory.Bicategory.whiskerLeft f.2
((CategoryTheory.Bicategory.leftUnitor g.1).prod (CategoryTheory.Bicategory.leftUnitor g.2)).hom.2)) =
CategoryTheory.Prod.mkHom
(CategoryTheory.Bicategory.whiskerRight
((CategoryTheory.Bicategory.rightUnitor f.1).prod (CategoryTheory.Bicategory.rightUnitor f.2)).hom.1 g.1)
(CategoryTheory.Bicategory.whiskerRight
((CategoryTheory.Bicategory.rightUnitor f.1).prod (CategoryTheory.Bicategory.rightUnitor f.2)).hom.2 g.2) |
AddCon.list_sum | Mathlib.GroupTheory.Congruence.BigOperators | ∀ {ι : Type u_1} {M : Type u_2} [inst : AddZeroClass M] (c : AddCon M) {l : List ι} {f g : ι → M},
(∀ x ∈ l, c (f x) (g x)) → c (List.map f l).sum (List.map g l).sum |
List.nil_eq_flatten_iff | Init.Data.List.Lemmas | ∀ {α : Type u_1} {L : List (List α)}, [] = L.flatten ↔ ∀ l ∈ L, l = [] |
_private.Lean.Meta.Tactic.Grind.Types.0.Lean.Meta.Grind.PendingSolverPropagationsData.rec | Lean.Meta.Tactic.Grind.Types | {motive : Lean.Meta.Grind.PendingSolverPropagationsData✝ → Sort u} →
motive Lean.Meta.Grind.PendingSolverPropagationsData.nil✝ →
((solverId : ℕ) →
(lhs rhs : Lean.Expr) →
(rest : Lean.Meta.Grind.PendingSolverPropagationsData✝¹) →
motive rest → motive (Lean.Meta.Grind.PendingSolverPropagationsData.eq✝ solverId lhs rhs rest)) →
((solverId : ℕ) →
(ps : Lean.Meta.Grind.ParentSet) →
(rest : Lean.Meta.Grind.PendingSolverPropagationsData✝²) →
motive rest → motive (Lean.Meta.Grind.PendingSolverPropagationsData.diseqs✝ solverId ps rest)) →
(t : Lean.Meta.Grind.PendingSolverPropagationsData✝³) → motive t |
_private.Lean.Meta.DiscrTree.Main.0.Lean.Meta.DiscrTree.reduceUntilBadKey.step._unsafe_rec | Lean.Meta.DiscrTree.Main | Lean.Expr → Lean.MetaM Lean.Expr |
Cardinal.mk_set_nat | Mathlib.SetTheory.Cardinal.Continuum | Cardinal.mk (Set ℕ) = Cardinal.continuum |
Submodule.comap_equiv_self_of_inj_of_le.match_1 | Mathlib.Algebra.Module.Submodule.Equiv | ∀ {R : Type u_2} {M : Type u_1} {N : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid N] [inst_4 : Module R N] {f : M →ₗ[R] N} {p : Submodule R N}
(motive : ↥(Submodule.comap f p) → Prop) (x : ↥(Submodule.comap f p)),
(∀ (val : M) (hx : val ∈ Submodule.comap f p), motive ⟨val, hx⟩) → motive x |
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