ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/WittVector/MulCoeff.lean
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n x : ℕ
hx : x ∈ range n
\u22a2 (↑p ^ x).vars ∪ (wittMul p x ^ p ^ (n - x)).vars ⊆ univ ×ˢ range n
-- Mathlib/RingTheory/WittVector/MulCoeff.lean
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n x : ℕ
hx : x ∈ range n
\u22a2 (↑p ^ x).vars ⊆ univ ×ˢ range n
-- Mathlib/RingTheory/WittVector/MulCoeff.lean
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n x : ℕ
hx : x ∈ range n
\u22a2 (wittMul p x ^ p ^ (n - x)).vars ⊆ univ ×ˢ range n
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
Q : AffineSubspace k P
\u22a2 Q = ⊥ ∨ (↑Q).Nonempty
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
Q : AffineSubspace k P
\u22a2 Q = ⊥ ∨ Q ≠ ⊥
-- Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
\u22a2 ((ofComplex.comp toComplex).toLinearMap ∘ₗ ι Q) 1 = ((AlgHom.id ℝ (CliffordAlgebra Q)).toLinearMap ∘ₗ ι Q) 1
-- Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
\u22a2 ofComplex (toComplex ((ι Q) 1)) = (ι Q) 1
-- Mathlib/Algebra/Category/Grp/EpiMono.lean
A B : Grp
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
\u22a2 (((Equiv.symm τ).trans (g x)).trans τ) ∞ = ∞
-- Mathlib/Algebra/Category/Grp/EpiMono.lean
A B : Grp
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
\u22a2 τ ((g x) ((Equiv.symm τ) ∞)) = ∞
-- Mathlib/Algebra/Category/Grp/EpiMono.lean
A B : Grp
f : A ⟶ B
x : ↑B
hx : x ∈ MonoidHom.range f
\u22a2 τ (fromCoset ⟨x • ↑⟨↑(MonoidHom.range f), ⋯⟩, ⋯⟩) = ∞
-- Mathlib/Data/Setoid/Partition.lean
α : Type u_1
r : Setoid α
a w✝ : α
ha : a ∈ {x | r.Rel x w✝}
x : α
\u22a2 x ∈ {x | r.Rel x w✝} ↔ x ∈ {x | r.Rel x a}
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
\u22a2 limsup (fun i => (xs i).toReal) F = (limsup xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
\u22a2 limsup (fun i => (xs i).toReal) F = (limsup xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (limsup xs F).toReal = b.truncateToReal (limsup xs F)
\u22a2 limsup (fun i => (xs i).toReal) F = (limsup xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (limsup xs F).toReal = b.truncateToReal (limsup xs F)
\u22a2 limsup (fun a => b.truncateToReal (xs a)) F = b.truncateToReal (limsup xs F)
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (limsup xs F).toReal = b.truncateToReal (limsup xs F)
key : b.truncateToReal (limsup xs F) = limsup (b.truncateToReal ∘ xs) F
\u22a2 limsup (fun a => b.truncateToReal (xs a)) F = b.truncateToReal (limsup xs F)
-- Mathlib/Probability/ProbabilityMassFunction/Monad.lean
α : Type u_1
β : Type u_2
γ : Type u_3
p : PMF α
f : α → PMF β
g : β → PMF γ
b : γ
\u22a2 ((p.bind f).bind g) b = (p.bind fun a => (f a).bind g) b
-- Mathlib/Tactic/NormNum/LegendreSymbol.lean
b : ℕ
hb : (b / 2).beq 0 = false
\u22a2 jacobiSymNat 0 b = 0
-- Mathlib/Tactic/NormNum/LegendreSymbol.lean
b : ℕ
hb : (b / 2).beq 0 = false
\u22a2 1 < b
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsLocalMinOn f (Icc a b) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
\u22a2 c ∈ {x | (fun x => f a ≤ f x) x}
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsLocalMinOn f (Icc a b) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
hc : a ∈ Icc a b
\u22a2 f a ≤ f a
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
H₁ : ∀ᶠ (y : ℝ) in 𝓝[>] a, f a ≤ f y
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
H₁ : ∀ᶠ (y : ℝ) in 𝓝[>] a, f a ≤ f y
H₂ : ∀ᶠ (y : ℝ) in 𝓝[>] a, y ∈ Ioc a c
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
H₁ : ∀ᶠ (y : ℝ) in 𝓝[>] a, f a ≤ f y
H₂ : ∀ᶠ (y : ℝ) in 𝓝[>] a, y ∈ Ioc a c
y : ℝ
hfy : f a ≤ f y
hy_ac : y ∈ Ioc a c
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
H₁ : ∀ᶠ (y : ℝ) in 𝓝[>] a, f a ≤ f y
H₂ : ∀ᶠ (y : ℝ) in 𝓝[>] a, y ∈ Ioc a c
ya yc : ℝ
ya₀ : 0 ≤ ya
yc₀ : 0 < yc
yac : ya + yc = 1
hfy : f a ≤ f (ya * a + yc * c)
hy_ac : ya * a + yc * c ∈ Ioc a c
\u22a2 f a ≤ f c
-- Mathlib/Analysis/Convex/Extrema.lean
E : Type u_1
β : Type u_2
inst✝⁷ : AddCommGroup E
inst✝⁶ : TopologicalSpace E
inst✝⁵ : Module ℝ E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module ℝ β
inst✝ : OrderedSMul ℝ β
s : Set E
f : ℝ → β
a b : ℝ
a_lt_b : a < b
h_local_min : IsMinFilter f (𝓝[≥] a) a
h_conv : ConvexOn ℝ (Icc a b) f
c : ℝ
hc : c ∈ Icc a b
a_lt_c : a < c
H₁ : ∀ᶠ (y : ℝ) in 𝓝[>] a, f a ≤ f y
H₂ : ∀ᶠ (y : ℝ) in 𝓝[>] a, y ∈ Ioc a c
ya yc : ℝ
ya₀ : 0 ≤ ya
yc₀ : 0 < yc
yac : ya + yc = 1
hfy : f a ≤ f (ya * a + yc * c)
hy_ac : ya * a + yc * c ∈ Ioc a c
\u22a2 ya • f a + yc • f a ≤ ya • f a + yc • f c
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
K : Type u_1
inst✝ : Field K
x : K
\u22a2 (commMap K) ((canonicalEmbedding K) x) = (mixedEmbedding K) x
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
K : Type u_1
inst✝ : Field K
x : K
\u22a2 (fun w => ((↑w).embedding x).re) = (Pi.ringHom fun w => embedding_of_isReal ⋯) x ∧ (fun w => (↑w).embedding x) = (Pi.ringHom fun w => (↑w).embedding) x
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
y : ℝ
hy : y ≠ 0
x : ℝ≥0∞
\u22a2 (x ^ y) ^ y⁻¹ = x
-- Mathlib/RingTheory/WittVector/WittPolynomial.lean
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
S : Type u_2
inst✝ : CommRing S
\u22a2 W_ R 0 = X 0
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
γ✝ γ : Path x y
γ' : Path y z
t : ↑I
\u22a2 (if ↑t ≤ 1 / 2 then γ.extend (2 * ↑t) else γ'.extend (2 * ↑t - 1)) = if h : ↑t ≤ 1 / 2 then γ ⟨2 * ↑t, ⋯⟩ else γ' ⟨2 * ↑t - 1, ⋯⟩
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s s₁ s₂ t t₁ t₂ u v : Finset α
a b : α
\u22a2 a ∉ s ∪ t ↔ a ∉ s ∧ a ∉ t
-- Mathlib/Data/Nat/Nth.lean
p : ℕ → Prop
inst✝ : DecidablePred p
k a : ℕ
ha : a ∈ filter p (range (nth p (k + 1)))
\u22a2 a ∈ insert (nth p k) (filter p (range (nth p k)))
-- Mathlib/Data/Nat/Nth.lean
p : ℕ → Prop
inst✝ : DecidablePred p
k a : ℕ
ha : a < nth p (k + 1) ∧ p a
\u22a2 a = nth p k ∨ a < nth p k ∧ p a
-- Mathlib/Data/Set/Pointwise/SMul.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Zero β
inst✝ : SMulZeroClass α β
s : Set α
t : Set β
a : α
hs : s.Nonempty
\u22a2 0 ⊆ s • 0
-- Mathlib/Data/Multiset/Bind.lean
α : Type u_1
β : Type v
γ : Type u_2
δ : Type u_3
f : α → β
S : Multiset (Multiset α)
\u22a2 map f S.join = (map (map f) S).join
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
\u22a2 mk ⁻¹' (mk '' s) = s
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x y z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
\u22a2 mk ⁻¹' (mk '' s) ⊆ s
-- Mathlib/Topology/Inseparable.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : (i : ι) → TopologicalSpace (π i)
x✝ y✝ z : X
s : Set X
f g : X → Y
t : Set (SeparationQuotient X)
hs : IsClosed s
x y : X
hys : y ∈ s
hxy : mk y = mk x
\u22a2 x ∈ s
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
z : (forget CommRingCat).obj (CommRingCat.of (A⁰_ f))
\u22a2 (awayToSection 𝒜 f ≫ (structureSheaf 𝒜).presheaf.germ x) z = (HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv) z
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
z : (forget CommRingCat).obj (CommRingCat.of (A⁰_ f))
\u22a2 (Proj.stalkIso' 𝒜 ↑x).toEquiv ((awayToSection 𝒜 f ≫ (structureSheaf 𝒜).presheaf.germ x) z) = (HomogeneousLocalization.mapId 𝒜 ⋯) z
-- Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
m : (i : Fin n.succ) → Ei i
\u22a2 f.curryRight.uncurryRight m = f m
-- .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean
a : Rat
\u22a2 a.num /. ↑a.den = a
-- Mathlib/RingTheory/Nilpotent/Defs.lean
R : Type u_1
S✝ : Type u_2
x✝ y : R
n : ℕ
S : Type u_3
inst✝ : MonoidWithZero S
x : S
N : ℕ
hN : x ^ N = 0
\u22a2 IsNilpotent (x ^ n.succ)
-- Mathlib/RingTheory/Nilpotent/Defs.lean
R : Type u_1
S✝ : Type u_2
x✝ y : R
n : ℕ
S : Type u_3
inst✝ : MonoidWithZero S
x : S
N : ℕ
hN : x ^ N = 0
\u22a2 (x ^ n.succ) ^ N = 0
-- Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : HasBinaryCoproduct X Y
\u22a2 coprod.inl ≫ (pushoutZeroZeroIso X Y).inv = pushout.inl 0 0
-- Mathlib/InformationTheory/Hamming.lean
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → DecidableEq (β i)
γ : ι → Type u_4
inst✝ : (i : ι) → DecidableEq (γ i)
x y : (i : ι) → β i
\u22a2 hammingDist x y < 1 ↔ x = y
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
\u22a2 ∃ g, ∀ (x : X), ‖↑g x - f x‖ < ε
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ‖↑g - { toFun := f, continuous_toFun := c }‖ < ε
\u22a2 ∃ g, ∀ (x : X), ‖↑g x - f x‖ < ε
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ‖↑g - { toFun := f, continuous_toFun := c }‖ < ε
\u22a2 ∀ (x : X), ‖↑g x - f x‖ < ε
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty α
inst✝ : One β
x : β
\u22a2 x ∈ SimpleFunc.range 1 ↔ x ∈ {1}
-- Mathlib/LinearAlgebra/FreeModule/PID.lean
ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
b : ι → M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
m : ↥N
i : Fin n
\u22a2 (⇑(snf.bM.repr ↑m) ∘ ⇑snf.f) i = (snf.a • ⇑(snf.bN.repr m)) i
-- Mathlib/RingTheory/WittVector/Compare.lean
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
m : ℕ
hm : n ≤ m
x : ZMod (p ^ m)
\u22a2 ((truncate hm).comp (zmodEquivTrunc p m).toRingHom) x = (zmodEquivTrunc p n) ((ZMod.castHom ⋯ (ZMod (p ^ n))) x)
-- Mathlib/RingTheory/WittVector/Compare.lean
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
m : ℕ
hm : n ≤ m
x : ZMod (p ^ m)
\u22a2 ((zmodEquivTrunc p n).toRingHom.comp (ZMod.castHom ⋯ (ZMod (p ^ n)))) x = (zmodEquivTrunc p n) ((ZMod.castHom ⋯ (ZMod (p ^ n))) x)
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
\u22a2 𝓝 1 = toFilterBasis.filter
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
\u22a2 map (fun y => y) toFilterBasis.filter = toFilterBasis.filter
-- Mathlib/Algebra/Associated/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a b c d : α
\u22a2 a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c
-- Mathlib/Algebra/Associated/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a b c d : α
\u22a2 b * a ~ᵤ d * c → b ~ᵤ d → b ≠ 0 → a ~ᵤ c
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
\u22a2 a ≠ b → a ∈ s → a ∈ s.erase b
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
\u22a2 a ≠ b → a ∈ s → a ≠ b ∧ a ∈ s
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
htμ : t ⟂ᵥ μ.toENNRealVectorMeasure
\u22a2 (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x)) ⟂ₘ t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
f : α → ℝ
hf : Measurable f
htμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ
\u22a2 (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x)) ⟂ₘ t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)
-- Mathlib/Topology/Homotopy/Product.lean
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
as bs cs : (i : ι) → X i
p : Homotopic.Quotient as bs
\u22a2 (pi fun i => proj i p) = p
-- Mathlib/Topology/Homotopy/Product.lean
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
as bs cs : (i : ι) → X i
a✝ : Path as bs
\u22a2 (pi fun i => proj i ⟦a✝⟧) = ⟦a✝⟧
-- Mathlib/Topology/Homotopy/Product.lean
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
as bs cs : (i : ι) → X i
a✝ : Path as bs
\u22a2 (pi fun i => ⟦a✝.map ⋯⟧) = ⟦a✝⟧
-- Mathlib/Topology/Homotopy/Product.lean
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
as bs cs : (i : ι) → X i
a✝ : Path as bs
\u22a2 ⟦Path.pi fun i => a✝.map ⋯⟧ = ⟦a✝⟧
-- Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
\u22a2 ((HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains A) 0 1 2 isoOneCocycles.proof_2 isoOneCocycles.proof_3).hom ≫ cyclesMap { τ₁ := ↑(zeroCochainsLequiv A), τ₂ := ↑(oneCochainsLequiv A), τ₃ := ↑(twoCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫ (shortComplexH1 A).moduleCatCyclesIso.hom) ≫ ModuleCat.ofHom (oneCocycles A).subtype = iCocycles A 1 ≫ ↑(oneCochainsLequiv A)
-- Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
\u22a2 (HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains A) 0 1 2 isoOneCocycles.proof_2 isoOneCocycles.proof_3).hom ≫ cyclesMap { τ₁ := ↑(zeroCochainsLequiv A), τ₂ := ↑(oneCochainsLequiv A), τ₃ := ↑(twoCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫ (shortComplexH1 A).moduleCatCyclesIso.hom ≫ ModuleCat.ofHom (oneCocycles A).subtype = iCocycles A 1 ≫ ↑(oneCochainsLequiv A)
-- Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
\u22a2 (HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains A) 0 1 2 isoOneCocycles.proof_2 isoOneCocycles.proof_3).hom ≫ cyclesMap { τ₁ := ↑(zeroCochainsLequiv A), τ₂ := ↑(oneCochainsLequiv A), τ₃ := ↑(twoCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫ (shortComplexH1 A).iCycles = iCocycles A 1 ≫ ↑(oneCochainsLequiv A)
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : AddMonoid G
x y : k
\u22a2 { toFun := (↑__src✝).toFun, map_one' := ⋯ }.toFun (x * y) = { toFun := (↑__src✝).toFun, map_one' := ⋯ }.toFun x * { toFun := (↑__src✝).toFun, map_one' := ⋯ }.toFun y
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
\u22a2 μ K = ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), μ U
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
\u22a2 μ K ≤ ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), μ U
-- Mathlib/MeasureTheory/Measure/Regular.lean
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
μ : Measure α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
K : Set α
hK : IsCompact K
\u22a2 ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), μ U ≤ μ K
-- Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Semiring B
inst✝² : Algebra R B
F : Type u_4
inst✝¹ : FunLike F A B
inst✝ : AlgHomClass F R A B
φ ψ : F
S T : Subalgebra R A
hS : Set.EqOn ⇑φ ⇑ψ ↑S
hT : Set.EqOn ⇑φ ⇑ψ ↑T
\u22a2 Set.EqOn ⇑φ ⇑ψ ↑(S ⊔ T)
-- Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
inst✝³ : Semiring B
inst✝² : Algebra R B
F : Type u_4
inst✝¹ : FunLike F A B
inst✝ : AlgHomClass F R A B
φ ψ : F
S T : Subalgebra R A
hS : S ≤ equalizer φ ψ
hT : T ≤ equalizer φ ψ
\u22a2 S ⊔ T ≤ equalizer φ ψ
-- Mathlib/Analysis/Calculus/FDeriv/Basic.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
\u22a2 DifferentiableOn 𝕜 f univ ↔ Differentiable 𝕜 f
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r✝ : ↑(Γ.obj (op X))
r s : (InducedCategory (Opens (PrimeSpectrum ↑(Γ.obj (op X)))) basicOpen)ᵒᵖ
f : r ⟶ s
\u22a2 ((inducedFunctor basicOpen).op ⋙ (structureSheaf ↑(Γ.obj (op X))).val).map f ≫ (fun r => X.toΓSpecCApp (unop r)) s = (fun r => X.toΓSpecCApp (unop r)) r ≫ ((inducedFunctor basicOpen).op ⋙ ((TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase).obj X.𝒪).val).map f
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r✝ : ↑(Γ.obj (op X))
r s : (InducedCategory (Opens (PrimeSpectrum ↑(Γ.obj (op X)))) basicOpen)ᵒᵖ
f : r ⟶ s
\u22a2 toOpen (↑(Γ.obj (op X))) (basicOpen (unop r)) ≫ ((inducedFunctor basicOpen).op ⋙ (structureSheaf ↑(Γ.obj (op X))).val).map f ≫ (fun r => X.toΓSpecCApp (unop r)) s = toOpen (↑(Γ.obj (op X))) (basicOpen (unop r)) ≫ (fun r => X.toΓSpecCApp (unop r)) r ≫ ((inducedFunctor basicOpen).op ⋙ ((TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase).obj X.𝒪).val).map f