ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
ho : lift.{v, u} o.card < c
\u22a2 lift.{v, u} o.card < c.ord.cof
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
\u22a2 ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
\u22a2 ‖y‖ * ‖y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
y z : E
h₀ : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
\u22a2 ‖y‖ * ‖y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
-- .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean
α : Type u
β : Type v
a : α
l : List α
\u22a2 (a :: l).eraseIdx 0 = a :: l ↔ (a :: l).length ≤ 0
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
X : Type u_3
Y : Type u_4
Z : Type u_5
β : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : StandardBorelSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : T0Space Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace Y
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace Z
inst✝ : CountablySeparated Z
f : X → Z
hf : Measurable f
hsurj : Surjective f
s : Set Z
h : MeasurableSet (f ⁻¹' s)
\u22a2 MeasurableSet s
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
X : Type u_3
Y : Type u_4
Z : Type u_5
β : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : StandardBorelSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : T0Space Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace Y
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace Z
inst✝ : CountablySeparated Z
f : X → Z
hf : Measurable f
hsurj : Surjective f
s : Set Z
h : MeasurableSet (f ⁻¹' s)
τ : TopologicalSpace Z
left✝¹ : SecondCountableTopology Z
left✝ : T4Space Z
right✝ : OpensMeasurableSpace Z
\u22a2 MeasurableSet s
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
X : Type u_3
Y : Type u_4
Z : Type u_5
β : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : StandardBorelSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : T0Space Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace Y
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace Z
inst✝ : CountablySeparated Z
f : X → Z
hf : Measurable f
hsurj : Surjective f
s : Set Z
h : MeasurableSet (f ⁻¹' s)
τ : TopologicalSpace Z
left✝¹ : SecondCountableTopology Z
left✝ : T4Space Z
right✝ : OpensMeasurableSpace Z
\u22a2 AnalyticSet s
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
X : Type u_3
Y : Type u_4
Z : Type u_5
β : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : StandardBorelSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : T0Space Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace Y
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace Z
inst✝ : CountablySeparated Z
f : X → Z
hf : Measurable f
hsurj : Surjective f
s : Set Z
h : MeasurableSet (f ⁻¹' s)
τ : TopologicalSpace Z
left✝¹ : SecondCountableTopology Z
left✝ : T4Space Z
right✝ : OpensMeasurableSpace Z
\u22a2 AnalyticSet sᶜ
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
\u22a2 levyProkhorovEDist μ ν ≤ δ
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
\u22a2 ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
\u22a2 ν B ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
\u22a2 ν B ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
\u22a2 ν (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
h : μ (thickening ε.toReal B)ᶜ ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
almost : 1 - μ (thickening ε.toReal B) + μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε + μ (thickening ε.toReal B)
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
almost : 1 ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + (ε + μ (thickening ε.toReal B))
\u22a2 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
almost : 1 ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + (ε + μ (thickening ε.toReal B))
\u22a2 ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + (ε + μ (thickening ε.toReal B)) - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean
α : Type u_1
β : Type u_2
𝕜 : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
f✝ g✝ : α → β
γ : Type u_4
inst✝ : TopologicalSpace γ
f : α → β
g : β → γ
hg : Continuous g
hf : AEStronglyMeasurable' m f μ
x : α
hx : f x = mk f hf x
\u22a2 (g ∘ f) x = (fun x => g (mk f hf x)) x
-- Mathlib/Analysis/NormedSpace/Real.lean
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
\u22a2 frontier (closedBall x r) = sphere x r
-- Mathlib/RingTheory/Coprime/Basic.lean
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime x y
H2 : IsCoprime x z
\u22a2 IsCoprime x (y * z)
-- Mathlib/RingTheory/Coprime/Basic.lean
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : IsCoprime y x
H2 : IsCoprime z x
\u22a2 IsCoprime (y * z) x
-- Mathlib/LinearAlgebra/LinearPMap.lean
R : Type u_1
inst✝⁹ : Ring R
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module R E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module R F
G : Type u_4
inst✝⁴ : AddCommGroup G
inst✝³ : Module R G
M : Type u_5
inst✝² : Monoid M
inst✝¹ : DistribMulAction M F
inst✝ : SMulCommClass R M F
y✝ : M
f : E →ₗ.[R] F
x : E
y : F
h : ∃ y_1, ↑y_1 = (x, y).1 ∧ ↑f y_1 = (x, y).2
\u22a2 x ∈ f.domain
-- Mathlib/LinearAlgebra/LinearPMap.lean
R : Type u_1
inst✝⁹ : Ring R
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module R E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module R F
G : Type u_4
inst✝⁴ : AddCommGroup G
inst✝³ : Module R G
M : Type u_5
inst✝² : Monoid M
inst✝¹ : DistribMulAction M F
inst✝ : SMulCommClass R M F
y✝ : M
f : E →ₗ.[R] F
x : E
y : F
x' : ↥f.domain
h : ↑x' = x ∧ ↑f x' = y
\u22a2 x ∈ f.domain
-- Mathlib/LinearAlgebra/LinearPMap.lean
R : Type u_1
inst✝⁹ : Ring R
E : Type u_2
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module R E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module R F
G : Type u_4
inst✝⁴ : AddCommGroup G
inst✝³ : Module R G
M : Type u_5
inst✝² : Monoid M
inst✝¹ : DistribMulAction M F
inst✝ : SMulCommClass R M F
y✝ : M
f : E →ₗ.[R] F
x : E
y : F
x' : ↥f.domain
h : ↑x' = x ∧ ↑f x' = y
\u22a2 ↑x' ∈ f.domain
-- Mathlib/Combinatorics/SetFamily/Compression/UV.lean
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
h : a ∈ 𝓒 u v s
ha : a ∉ s
\u22a2 Disjoint v a
-- Mathlib/Combinatorics/SetFamily/Compression/UV.lean
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s : Finset α
u v a b : α
inst✝ : DecidableEq α
h : a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a
ha : a ∉ s
\u22a2 Disjoint v a
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
\u22a2 @tails = @tailsTR
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
\u22a2 tails = tailsTR
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
h : a₁ • b₁ = a₂ • b₂
\u22a2 a₁ = a₂ ∧ b₁ = b₂
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
h : a₁ • b₁ = a₂ • b₂
\u22a2 ¬a₁ < a₂ ∧ ¬b₁ < b₂
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha✝ : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
h : a₁ • b₁ = a₂ • b₂
ha : a₁ < a₂
\u22a2 a₁ • b₁ < a₂ • b₂
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb✝ : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
h : a₁ • b₁ = a₂ • b₂
hb : b₁ < b₂
\u22a2 a₁ • b₁ < a₂ • b₂
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions/ComposableArrows.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
n : ℕ
\u22a2 (L.mapComposableArrows n).EssSurj
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions/ComposableArrows.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
n : ℕ
this : (L.op.mapComposableArrows n).EssSurj
\u22a2 (L.mapComposableArrows n).EssSurj
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions/ComposableArrows.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
n : ℕ
this✝ : (L.op.mapComposableArrows n).EssSurj
this : (L.mapComposableArrows n ⋙ (opEquivalence D n).functor.rightOp).EssSurj
\u22a2 (L.mapComposableArrows n).EssSurj
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → (Matroid.sigma M).Indep J
\u22a2 (Matroid.sigma M).Indep I
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
\u22a2 ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
i : ι
\u22a2 (M i).Indep (Sigma.mk i ⁻¹' I)
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
i : ι
\u22a2 ∀ J ⊆ Sigma.mk i ⁻¹' I, J.Finite → (M i).Indep J
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
i : ι
J : Set (α i)
hJI : J ⊆ Sigma.mk i ⁻¹' I
hJ : J.Finite
\u22a2 (M i).Indep J
-- Mathlib/Data/Matroid/Sum.lean
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
h : ∀ (i : ι), (M i).Finitary
I : Set ((i : ι) × α i)
hI : ∀ J ⊆ I, J.Finite → ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' J)
i : ι
J : Set (α i)
hJI : J ⊆ Sigma.mk i ⁻¹' I
hJ : J.Finite
\u22a2 J = Sigma.mk i ⁻¹' (Sigma.mk i '' J)
-- Mathlib/Topology/Algebra/Module/Basic.lean
R : Type u_1
R₂ : Type u_2
R₃ : Type u_3
S : Type u_4
S₃ : Type u_5
inst✝³⁶ : Semiring R
inst✝³⁵ : Semiring R₂
inst✝³⁴ : Semiring R₃
inst✝³³ : Monoid S
inst✝³² : Monoid S₃
M : Type u_6
inst✝³¹ : TopologicalSpace M
inst✝³⁰ : AddCommMonoid M
inst✝²⁹ : Module R M
M₂ : Type u_7
inst✝²⁸ : TopologicalSpace M₂
inst✝²⁷ : AddCommMonoid M₂
inst✝²⁶ : Module R₂ M₂
M₃ : Type u_8
inst✝²⁵ : TopologicalSpace M₃
inst✝²⁴ : AddCommMonoid M₃
inst✝²³ : Module R₃ M₃
N₂ : Type u_9
inst✝²² : TopologicalSpace N₂
inst✝²¹ : AddCommMonoid N₂
inst✝²⁰ : Module R N₂
N₃ : Type u_10
inst✝¹⁹ : TopologicalSpace N₃
inst✝¹⁸ : AddCommMonoid N₃
inst✝¹⁷ : Module R N₃
inst✝¹⁶ : DistribMulAction S₃ M₃
inst✝¹⁵ : SMulCommClass R₃ S₃ M₃
inst✝¹⁴ : ContinuousConstSMul S₃ M₃
inst✝¹³ : DistribMulAction S N₃
inst✝¹² : SMulCommClass R S N₃
inst✝¹¹ : ContinuousConstSMul S N₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝⁹ : DistribMulAction S₃ M₂
inst✝⁸ : ContinuousConstSMul S₃ M₂
inst✝⁷ : SMulCommClass R₂ S₃ M₂
inst✝⁶ : DistribMulAction S N₂
inst✝⁵ : ContinuousConstSMul S N₂
inst✝⁴ : SMulCommClass R S N₂
inst✝³ : SMulCommClass R₂ R₂ M₂
inst✝² : SMulCommClass R₃ R₃ M₃
inst✝¹ : ContinuousConstSMul R₂ M₂
inst✝ : ContinuousConstSMul R₃ M₃
h : M₂ →SL[σ₂₃] M₃
c : R₂
f : M →SL[σ₁₂] M₂
x : M
\u22a2 (h.comp (c • f)) x = (σ₂₃ c • h.comp f) x
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommMonoid S
inst✝ : DistribMulAction S M
x y : M
a : S
hx : a • x = 0
b : S
hy : b • y = 0
\u22a2 x + y ∈ {x | ∃ a, a • x = 0}
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_3
inst✝¹ : CommMonoid S
inst✝ : DistribMulAction S M
x y : M
a : S
hx : a • x = 0
b : S
hy : b • y = 0
\u22a2 (b * a) • (x + y) = 0
-- Mathlib/Algebra/GCDMonoid/Multiset.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
inst✝ : DecidableEq α
s✝ : Multiset α
a : α
s : Multiset α
IH : s.dedup.lcm = s.lcm
h : a ∈ s
\u22a2 fold GCDMonoid.lcm 1 s = GCDMonoid.lcm a (fold GCDMonoid.lcm 1 s)
-- Mathlib/Algebra/GCDMonoid/Multiset.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
inst✝ : DecidableEq α
s✝ : Multiset α
a : α
s : Multiset α
IH : s.dedup.lcm = s.lcm
h : a ∈ s
\u22a2 GCDMonoid.lcm a (fold GCDMonoid.lcm 1 (s.erase a)) = GCDMonoid.lcm (normalize a) (fold GCDMonoid.lcm 1 (s.erase a))
-- Mathlib/FieldTheory/KummerExtension.lean
K : Type u
inst✝³ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
hn : 0 < n
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
\u22a2 adjoin K {α} = ⊤
-- Mathlib/FieldTheory/KummerExtension.lean
K : Type u
inst✝³ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
hn : 0 < n
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
\u22a2 Subalgebra.map (↑(adjoinRootXPowSubCEquiv hζ H hα).symm) (adjoin K {α}) = Subalgebra.map ↑(adjoinRootXPowSubCEquiv hζ H hα).symm ⊤
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : Lattice α
inst✝ : OrderBot α
a : α
P : Finpartition a
\u22a2 P.parts = ∅ ↔ a = ⊥
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : Lattice α
inst✝ : OrderBot α
a : α
P : Finpartition a
\u22a2 P.parts = ∅ ↔ P.parts.sup id = ⊥
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : Lattice α
inst✝ : OrderBot α
a : α
P : Finpartition a
h : P.parts = ∅
\u22a2 P.parts.sup id = ⊥
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : Lattice α
inst✝ : OrderBot α
a : α
P : Finpartition a
h : P.parts.sup id = ⊥
b : α
hb : b ∈ P.parts
\u22a2 ⊥ ∈ P.parts
-- Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
n : ℕ
hn : n % 4 = 3
\u22a2 χ₄ ↑n = -1
-- Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
n : ℕ
hn : n % 4 = 3
\u22a2 χ₄ ↑3 = -1
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝⁶ : SemilatticeSup α
inst✝⁵ : OrderTop α
inst✝⁴ : DecidableRel fun x x_1 => x ≤ x_1
inst✝³ : SemilatticeSup β
inst✝² : BoundedOrder β
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a b : α
inst✝ : DecidableEq α
hs : a ∈ lowerClosure ↑s
ht : a ∉ lowerClosure ↑t
\u22a2 (s ∪ t).truncatedSup a = s.truncatedSup a
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝⁶ : SemilatticeSup α
inst✝⁵ : OrderTop α
inst✝⁴ : DecidableRel fun x x_1 => x ≤ x_1
inst✝³ : SemilatticeSup β
inst✝² : BoundedOrder β
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a b : α
inst✝ : DecidableEq α
hs : a ∈ lowerClosure ↑s
ht : ∀ x ∈ t, ¬a ≤ x
\u22a2 (s ∪ t).truncatedSup a = s.truncatedSup a
-- Mathlib/Probability/Distributions/Exponential.lean
r x : ℝ
\u22a2 ENNReal.ofReal (if 0 ≤ x then r ^ 1 / Gamma 1 * x ^ (1 - 1) * rexp (-(r * x)) else 0) = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)
-- Mathlib/Dynamics/OmegaLimit.lean
τ : Type u_1
inst✝³ : TopologicalSpace τ
inst✝² : AddCommGroup τ
inst✝¹ : TopologicalAddGroup τ
α : Type u_2
inst✝ : TopologicalSpace α
f : Filter τ
ϕ : Flow τ α
s : Set α
hf : ∀ (t : τ), Tendsto (fun x => x + t) f f
t : τ
\u22a2 ω f ϕ.toFun s = ω f ϕ.toFun (ϕ.toFun (-t) '' (ϕ.toFun t '' s))
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hx : x₁ = x₂
\u22a2 y₁ = y₂ ∨ y₁ = W.negY x₂ y₂
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : y₁ ^ 2 + W.a₁ * x₁ * y₁ + W.a₃ * y₁ = x₁ ^ 3 + W.a₂ * x₁ ^ 2 + W.a₄ * x₁ + W.a₆
h₂ : y₂ ^ 2 + W.a₁ * x₂ * y₂ + W.a₃ * y₂ = x₂ ^ 3 + W.a₂ * x₂ ^ 2 + W.a₄ * x₂ + W.a₆
hx : x₁ = x₂
\u22a2 y₁ = y₂ ∨ y₁ = W.negY x₂ y₂